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A Comprehensive  Arithmetic 

v 

FOR 


GRAMMAR,  HIGH  and  COMMERCIAL  SCHOOLS 


REVISED  AND  EDITED  BY 

JOHN  A.  LUMAN,  A.  M. 

Vice-Principal  of  Peirce  School 


FOURTH  EDITION 


PHILADELPHIA  : 

Published  by  PEIRCE  SCHOOL 

Nos.  9 1 7-9  1 9 Chestnut  Street 
I 908 


COPYRIGHT,  1 908 

BY 

PEIRCE  SCHOOL 


T.  C.  DAVIS  & SONS.  PRINTERS.  PH1LA 


\ p-|q \s  =- 

AT.  “CH- 

J5D 


5 II 

X^I 


PREFACE 


IN  the  preparation  of  this  book,  great  care  and  attention  have  been  given  to 
the  logical  relation  of  the  science,  and  its  adaptation  to  practical  affairs  ; 
to  definiteness  and  exactness  in  principles  and  rules  ; to  brevity  and  clearness 
in  the  presentation  of  the  subject  matter ; and  to  modern  progressive  methods. 

The  aim  has  been  to  present  the  truths  and  principles  of  a practical 
science  in  a practical,  business-like  way,  and  to  develop  them  in  a natural, 
simple,  effective  manner,  with  a variety  of  well-selected  problems  of  a character 
suited  to  students  of  grammar,  high  and  commercial  schools. 

The  features  of  this  book  are  : 

i — The  logical  development  of  number. 

( a ) The  properties  of  number. 

(<£)  The  relation  of  a quantity  to  a fixed  unit  of  measure. 

(<r)  The  mechanical  and  reasoning  processes  by  analysis,  comparison 


2 —  The  definitions  of  terms  in  clear,  concise,  arithmetical  language. 

3 —  The  fundamental  operations  made  simple  and  easy  by  short  processes. 

4 —  Models  in  script,  with  illustrative  figures,  giving  specific  concrete  form  to 

the  principles  discussed. 

5 —  A full  treatment  of  common  and  decimal  fractions,  including  a new  and 

safe  method  of  decimal  division. 

6 —  The  logical  development  of  measurements,  copiously  illustrated  by  cuts 

and  figures,  and  by  problems  from  practical  life. 

7 —  A short  and  novel  presentation  of  the  operations  of  percentage,  trade  dis- 

count, commission,  interest,  and  the  metric  system. 

8 —  The  comprehensive  treatment  of  exchange,  partnership  and  partnership 

settlements,  averaging  and  equating  accounts,  stocks  and  bonds,  men- 
suration, etc. 

9 -The  classification  and  gradation  of  problems — first,  simple  mental  prob- 
lems to  develop  an  insight  into  the  principles ; second,  plain  ques- 
tions in  calculation  to  develop  ease  and  quickness ; third,  more  com- 
plicated problems  requiring  reasoning  as  well  as  computation, 
io — The  introduction  of  many  interesting  and  helpful  notes. 

No  effort  has  been  spared  to  write  a simple,  logical,  comprehensive 
book  on  arithmetic  that  meets  the  needs  of  the  student  and  teacher  in  the  class- 
room, as  well  as  the  demands  of  present-day  industrial  and  commercial  life. 

Appreciation  and  thanks  are  extended  to  J.  H.  Bryant,  F.  H.  Hain, 
E.  J.  Conner  and  R.  F.  Hayes  for  suggestions  and  help;  to  R.  S.  Collins  for 
script  work  ; and  to  Thomas  May  Peirce,  Jr.,  for  illustrative  figures. 


and  synthesis. 


CONTENTS 


Definitions  and  Principles 

Notation  and  Numeration 

Addition  

Cross  Addition  and  Tabular  Work  . . 

Subtraction  

Cashiers’  Calculations 

Multiplication 

Salesmen’s  and  Cashiers’  Calculations 
Short  Methods  in  Multiplication  . . . 

Cross  Multiplication 

Sliding  Method  of  Multiplication  . . . 

Division 

Long  Division 

Short  Rules  in  Division 

Properties  of  Numbers 

Factoring  

Greatest  Common  Divisor 

Least  Common  Multiple 

Cancelation 

Common  Fractions 

Reduction  of  Fractions 

Addition  of  Fractions . . 

Subtraction  of  Fractions  

Multiplication  of  Fractions 

Division  of  Fractions • 

Complex  Fra  ctions  

Relation  of  Numbers 

Problems  in  Fractions 

Decimals 

Numeration  of  Decimals 

Notation  of  Decimals 

Reduction  of  Decimals 

Addition  of  Decimals 

Subtraction  of  Decimals 

Multiplication  of  Decimals 

Division  of  Decimals  . 

Short  Methods  in  Decimals  

Review  Exercise  in  Decimals 

Review  Problems 

Quantity,  Price  and  Cost 


PAGE 

I 

8 

11 

16 

18 

20 

24 

28 

29 

31 

33 

35 

38 

39 

43 

44 

45 

47 

48 

50 

51 
53 
56 
53 
60 
64 
66 
69 

74 

75 

76 

77 

79 

80 
81 
82 

85 

86 
88 
90 


4 


CONTENTS 


5 


PAGE 

Aliquot  Parts 91 

Special  Rules 94 

Analysis 98 

Measurements 105 

Denominate  Numbers 110 

Practical  Measurements 117 

Calculating  Lumber 127 

Approximate  Measurements 137 

Percentage 141 

Profit  and  Loss 152 

Invoice  Extension  and  Trade  Discount 163 

Commission  and  Brokerage 171 

Insurance 181 

Review  Problems  in  Percentage 184 

Interest 188 

Six  Per  Cent.  Method 192 

Day  Method 193 

60- Day  Method 194 

“$6000  Rule” 195 

Accurate  Interest 198 

Interest  Problems 199 

Compound  Interest • • 203 

Review  Problems  in  Interest 208 

Bank  Discount 209 

Partial  Payments 218 

United  States  Rule 218 

Mercantile  Rule 219 

Stocks  and  Bonds 222 

Stocks • 224 

Buying  and  Selling  on  Margin 226 

Buying  and  Selling  Short 227 

Bonds > 228 

Exchange 234 

Foreign  Exchange 237 

Taxes 240 

Duties  ...  243 

Ratio  and  Proportion 245 

Ratio 245 

Proportion  247 

Compound  Proportion 250 

Alligation 253 

General  Average 257 

Equation  of  Accounts 261 

Product  Method  263 

223632 


6 


CONTENTS 


PAGE 

Interest  Method 264 

Cash  Balance 269 

Accounts  Bearing  Interest 278 

Savings  Fund  Accounts 280 

Bankruptcy 282 

Partnership  Settlements 283 

Involution  and  Evolution  . 298 

Involution  298 

Evolution 299 

Square  Root  300 

Similar  Figures  301 

Cube  Root 302 

Similar  Solids  306 

Arithmetical  Progression 307 

Geometrical  Progression 309 

Mensuration  311 

Right-angled  Triangles 311 

Surfaces  313 

Solids  . . 318 

Latitude  and  Longitude 324 

Longitude  and  Time 325 

Review  Mental  Problems  328 

General  Review  Problems . 335 

Metric  System ...  361 


ARITHMETIC 

DEFINITIONS  AND  PRINCIPLES 

1.  Quantity  is  a limited  portion  of  any  natural  object;  as  of  time,  space, 
weight,  etc.,  or  of  any  solid  or  fluid  substance. 

2.  All  mathematical  operations  deal  with  measures  of  quantity , and  quantity 
can  be  measured  only  by  comparing  it  with  some  known  quantity  of  the  same 
kind  taken  as  a standard. 

3.  Quantity  can  be  changed,  mathematically,  only  by  increase  or  diminution. 

4.  A quantity  may  be  increased  in  either  of  two  ways:  (a)  By  combining 
with  it  one  or  more  quantities  of  the  same  kind,  greater  or  less  than  itself ; this 
is  called  Addition,  (b)  By  combining  with  it  any  given  number  of  quantities 
each  exactly  equal  to  itself ; this  is  called  Multiplication. 

5.  A quantity  may  be  diminished  in  either  of  two  ways  : (a)  By  taking 
from  it  a quantity  not  greater  than  itself ; this  is  called  Subtraction,  (b)  By 
continuing  to  take  from  it,  as  many  times  as  possible,  a given  quantity  less  than 
itself ; this  is  called  Division  (because  the  quantity  is  separated  into  a number  of 
equal  parts,  with  or  without  a remainder). 

6.  The  two  mathematical  principles,  increase  and  diminution , thus  give  rise 
to  the  four  fundamental  operations  of  arithmetic — Addition,  Subtraction,  Multi- 
plication and  Division. 

7.  Arithmetic,  as  a science,  is  the  science  of  numbers.  As  an  art,  it 
embraces  all  known  methods  of  computation  by  means  of  figures,  all  of  which 
are  but  processes  or  combinations  of  adding,  subtracting,  multiplying  and 
dividing. 

8.  The  Signs  of  the  four  cardinal  operations  are  as  follows  : Of  Addition, 
+ plus;  of  Subtraction,  — minus  ; of  Multiplication,  X multiplied  by  ; of  Divi- 
sion, -r-  divided  by. 

9.  The  Result  in  addition  is  called  the  sum;  in  subtraction,  the  difference 
or  remainder  ; in  multiplication,  the  product]  and  in  division,  the  quotient. 

10.  The  Sign  of  Equality  is  =,  read  equals  or  equal  to. 

11.  A Unit,  represented  by  the  figure  1,  signifies  one  thing  of  any  kind ; 
as,  1 dollar,  1 yard,  1 day.  When  not  applied  to  any  thing,  it  represents  the 
abstract  idea  of  unity. 

12.  A Number  is  a unit  or  a group  of  units,  expressed  by  one  or  more  figures, 
and  considered  as  one  quantity.  Numbers  may  be  abstract,  as  1,  5,  12  ; or  concrete, 
as  6 feet,  25  barrels. 


7 


NOTATION  AND  NUMERATION 


13.  Notation  is  the  art  of  writing  numbers.  For  purposes  of  calculation 
the  Arabic  nofation,  expressing  numbers  in  figures,  is  used  exclusively;  but  the 
Roman  notation,  a system  of  representing  numbers  by  combinations  of  letters,  is 
still  used  for  numbering  chapters,  dates  of  imprint  and  inscriptions,  dials,  etc. 

14.  Numeration  is  the  art  of  reading  or  naming  numbers. 

15.  Arabic  Notation  expresses  numbers  by  the  following  ten  characters : 
1,  one;  2,  two;  3,  three;  4,  four;  5,  five;  6,  six;  7,  seven;  8,  eight;  9,  nine; 
0,  naught,  cipher  or  zero.  They  are  written  as  follows  : 


Any  one  of  these  ten  characters  or  figures  is  called  a digit,  and  a whole 
number  represented  by  any  one  of  them  an  integer. 

The  number  of  units  expressed  by  any  one  of  these  figures  when  standing  alone  is  shown  by  its 
name  ; but  when  used  in  combination  with  other  figures  the  number  of  units  expressed  is  indicated  by 
its  place,  as  shown  in  the  table  which  follows.  A figure  occupying  any  one  of  these  places  expresses  as 
many  times  the  number  of  units  it  names  as  is  indicated  by  the  name  of  the  place  it  occupies,  being  in 
each  case  ten  times  as  many  as  it  would  represent  in  the  next  place  to  the  right.  Thus  the  figure  8,  with 
one  figure  at  the  right  of  it — as  in  80,  82,  or  86 — represents  8 tens,  because  it  occupies  the  second,  or 
tens  place  ; with  two  figures  following  to  the  right— as  in  800,  805,  or  837 — it  represents  8 hundreds, 
because  it  occupies  the  third,  or  hundreds  place,  etc.  The  cipher,  by  itself,  represents  nothing,  and  is 
used  merely  to  fill  vacant  places. 


16. 


NUMERATION  TABLE 


NAMES 

OF 

PLACES 


FLACES, 

OR 

ORDERS 


P o 

<L> 

HQ 


<D  •+=>  o <D  C_., 

o .2  u.  aT- 


S3  ^ 
r-» 


Who 


* w 


’■4^  fl 

x o 

& 


m 


^ 'Q  § 

« M • S 

t-«  <i)  <—* 

® <U  2 g"  <1>  a 

SbO1 


XU 

Oi  ZZ 
t-  3- 

= £.2 


: c3 
> 3 

1C* 


> .33  c 

: sm  o 


;hh 


; c ==: 

; a;  .— 

; h q 


p 


S S2  £ Q p g 2 c3  t 


<D  ^ 


10,  9 9 9,888,  777,  666,  555,444,  333,  222, 1 1 1,654,32  1 


rP  rP  _ _ -PrPrP 

“ Tj  (O 

COOf  H O O 00 
COCO  COCOCO  COCKM 


5 

CD  lO 
(M  <M  <M 


Q p: 

rr  CO  Cl 


hoc; 

Of  O*  WH 


LC  TP  CO  P?  ' 


O 5 33  33  'd 

rH  CiCCt^  C r CO  PJ  — 


PERIODS  12th  11th  10th 


9th 


8th 


7th 


6th 


5 th  4tli 


3d 


2d 


1st 


NAMES 

OF 

PERIODS 


o 

O) 

a 


a 

o 

& 


8 


NOTATION  AND  NUMERATION 


9 


17.  Dictionaries  of  the  English  language  do  not  contain  the  names  for  the 
periods  above  Decillions,  because  they  are  so  seldom  used.  It  will  be  observed 
that,  commencing  with  Millions  (in  lieu  of  Unillions)  as  one,  Billions  two,  Trillions 
three,  etc.,  these  names  consist  of  the  Latin  numerals  in  order,  with  the  termina- 
tion -illions. 

Note. — The  names  of  the  higher  periods,  sometimes  nsed  by  mathematicians,  are  formed  in  the 
same  way  ; but  that  snch  numbers  are  far  beyond  human  comprehension  may  be  readily  shown  by  the 
following  : If  a hollow  sphere  of  the  size  of  the  Earth  (say,  8000  miles  in  diameter)  were  filled  with 

wheat,  the  number  of  grains  in  the  entire  mass  would  be  between  thirteen  and  fourteen  octillions 
(allowing  200  grains  to  the  cubic  inch). 

18.  It  will  also  be  observed  that  there  is  a difference  of  two  between  the 
number  expressed  by  the  root  of  the  period-name  and  the  number  of  the  period 
itself — Octillions  ( odo , eight)  being  the  tenth  period,  for  instance — just  as  there  is 
a difference  of  two  between  the  numbers  indicated  by  the  roots  of  September, 
October,  November,  December,  and  the  numbers  of  these  months  in  our  year. 
The  reason  for  this  discrepancy  in  the  case  of  the  period-names  is  that  they  start 
with  Millions,  while  with  regard  to  the  month-names  mentioned  it  originates  in 
the  fact  that  the  Roman  year  began  with  March. 

19.  English  Numeration.  The  system  of  numeration  shown  in  the 
Numeration  Table  (Art.  16)  is  called  the  French  and  American  system;  it  is 
used  in  nearly  all  countries,  England  being  the  most  important  exception.  The 
English  method  is  the  same  as  far  as  Millions  ; beyond  Millions  the  names  are  as 
follows  : 


Numerals 

American  and  French 
Name 

English  Name 

1,000,000,000 

1012 

1015 

1018 

1021 

1024 

1027 

1030 

1033 

Billions 

Trillions 

Quadrillions 

Quintillions 

Sextillions 

Septillions 

Octillions 

Nonillions 

Decillions 

Thousand  millions 
Billions 

Thousand  billions 
Trillions 

Thousand  trillions 
Quadrillions 
Thousand  quadrillions 
Quintillions 
Thousand  quintillions 

In  the  French  and  American  system,  each  succeeding  period  of  three  figures 
has  a new  name,  so  that  each  period-name  indicates  a number  one  thousand  times 
as  great  as  the  name  preceding.  In  the  English  system,  the  figures  are  grouped 
in  periods  of  six  figures  each — Units,  Tens,  Hundreds,  Thousands,  Ten-thousands, 
Hundred-thousands — Millions,  Ten-millions,  Hundred-millions,  Thousand-mill- 
ions, Ten-thousand-millions,  Hundred-thousand-millions — Billions,  etc. — so  that 
each  period-name  indicates  a number  one  million  times  as  great  as  the  name  pre- 
ceding. 


10 


NOTATION  AND  NUMERATION 


20.  Roman  Notation.  The  Roman  notation  expresses  numbers  by  means 
of  seven  capital  letters,  as  follows:  I,  one;  V,  five;  X,  ten  ; L,  fifty  ; C.  one  hun- 
dred ; D,  five  hundred ; M,  one  thousand. 


ROMAN  NOTATION  TABLE 


Arabic 

Roman 

Arabic 

Koman 

Arabic 

Roman 

1 

I 

16 

XVI 

400 

CD 

2 

II 

17 

XVII 

500 

D 

3 

III 

18 

XVIII 

600 

DC 

4 

IV 

19 

XIX 

700 

DC’C 

5 

V 

20 

XX 

800 

DC’CC 

6 

VI 

30 

XXX 

900 

CM 

7 

VII 

40 

XL 

1,000 

M 

8 

VIII 

50 

L 

1,300 

MCCC 

9 

IX 

60 

LX 

1,400 

MCD 

10 

X 

70 

LXX 

1,500 

MD 

11 

XI 

80 

LXXX 

1,600 

MDC 

12 

XII 

90 

XC 

1,800 

MDCCC 

13 

XIII 

100 

c 

1,900 

MCM 

14 

XIV 

O 

O 

CM 

CC 

2,000 

MM 

15 

XV 

300 

CCC 

1,000,000 

M 

The  principles  by  which  the  letters  are  combined  to  express  numbers,  as  illustrated  in  the  fore- 
going table,  are  as  follows  : 

1.  The  letters  I,  X,  C and  M are  written  twice  or  three  times  in  succession  to  indicate  double  or 
triple  their  values  ; as  II,  two,  III,  three  ; XX,  twenty,  XXX,  thirty  ; CC,  two  hundred,  CCC,  three- 
hundred  ; MM,  two  thousand,  MMM,  three  thousand  ; but  they  are  not  repeated  more  than  three  times, 
because  there  is  a shorter  way  of  representation  by  Principle  2.  The  other  letters  are  not  repeated, 
because  VV=X,  LL=C,  and  DD=M. 

2.  When  a letter  of  less  value  is  placed  before  one  of  greater  value,  the  combination  represents 
the  difference  of  their  values,  i.  e.,  the  greater  minus  the  less  ; as,  IV,  four  ; IX,  nine  ; XL.  forty  ; XC. 
ninety  ; CD,  four  hundred  ; CM,  nine  hundred.  A letter  of  less  value  placed  between  two  of  greater 
value  is  thus  subtracted  from  their  sum  ; as,  XIV,  fourteen  ; XIX,  nineteen  ; CXL.  one  hundred  fortv; 
CXC,  one  hundred  ninety  ; MCM,  nineteen  hundred. 

3.  When  a letter  of  less  value  is  placed  after  one  of  greater  value,  the  combination  represents  the 
sum  of  their  values  ; as  VI,  six  ; XI,  eleven  ; LV,  fifty-five  ; DL,  five  hundred  fifty. 

4.  A line  (dash,  bar,  or  vinculum)  placed  over  a letter  or  letters  increases  the  value  one  thousand 
times ; as  IV,  four  thousand  ; X,  ten  thousand  ; LXV,  sixty-five  thousand  ; MM,  two  million. 

Note. — On  the  dials  of  watches  and  clocks  IIII  is  invariably  to  be  seen  in  place  of  IV,  the 
correct  form.  Custom  has  perpetuated  the  error.  Also,  the  date  of  imprint,  1900.  may  be  seen  on 
some  books  represented  by  MDCCCC  ; but  the  correct  form  is  MCM,  by  Principle  2. 


ADDITION 


21.  Addition  is  the  process  of  finding  the  sum  of  two  or  more  numbers. 

22.  The  Sum  or  Amount  of  several  numbers  is  one  number  which  con- 
tains as  many  units  as  there  are  in  all  the  numbers  added. 

23.  The  Sign  of  Addition  is  + , read  plus,  and  denotes  that  the  numbers 
between  which  it  is  placed  are  to  be  added. 

24.  The  Sign  of  Equality  is  =,  read  equals  or  equal  to,  and  denotes  that 
the  numbers  or  quantities  between  which  it  is  placed  are  equal. 

25.  The  Principles  of  Addition  are: 

1.  Only  like  numbers  can  be  added. 

2.  Units  of  the  same  order  only  can  be  directly  combined. 

3.  The  sum  is  a number  of  the  same  kind  as  the  numbers  added 

f.  The  sum  is  the  same  in  whatever  order  the  numbers  may  be  added. 


26.  To  find  the  sum  of  two  or  more  numbers. 

Example. — Add  328,  1459,  67  and  2904. 


Write  the  first  number,  then  place  the  others  beneath  it  in  such  a manner  that 
the  units  figures  shall  be  directly  under  units,  tens  under  tens,  hundreds  under 
hundreds,  etc. — being  careful  to  keep  the  columns  vertical  and  evenly  spaced. 
Beginning  with  the  units  (first  column  at  the  right)  add  4-f-7+9-f-8,  naming  result 
only  at  each  step  thus  : “ Eleven,  twenty — eight  ” (28),  which  gives  2 tens  and  8 
units.  (Do  not  say,  “Four  and  7 are  11,  and  9 are  20,  and  8 are  28.”)  Write  8 
under  the  units  column  and  carry  2 to  the  tens  column.  Add  the  tens  column  by 
combining  tbe  2 (carried)  with  the  6,  and  the  5 with  the  2,  saying,  “Eight — 
fifteen,”  which  gives  5 tens  and  1 hundred  to  carry.  Write  5 under  the  tens 
column,  and  carry  1 to  the  hundreds  column.  Add  the  hundreds  column  by  combining  the  1 (carried) 
with  the  9,  and  the  4 with  the  3,  saying,  “Ten,  seventeen” — 1 thousand  and  7 hundreds.  Write  7 
under  the  hundreds  column  and  carry  1 to  the  thousands  column,  which  is  now  read  as  “ Four,”  and 
the  4 written  under  it.  Result,  4758 — the  sum. 


3 z r 
/ v s? 

7 

2 7 0 4 
^ 7 ^ f 


27.  Rule. — Write  the  numbers  so  that  units  of  the  same  order  stand  in  the  same 
column. 

Beginning  with  the  units  column , add  each  column  separately,  and  write  the  sum 
beneath  it,  if  less  than  ten. 

If  the  sum  of  any  column  is  ten,  or  more  than  ten,  set  down  only  the  right-hand 
figure  under  the  column  added , and  add  the  remaining  figure  or  figures  to  the  next 
column. 

Write  the  entire  sum  of  the  last  column. 


28.  Proof. — Add  tbe  columns  again,  reversing  the  order  of  the  operation  ; 
that  is  to  say,  if  tbe  first  adding  has  been  from  bottom  of  column  to  top,  let  the 
second  be  from  top  of  column  to  bottom,  or  vice  versa. 


ll 


12 


ADDITION 


Note. — In  thus  reversing  the  order  of  operation,  new  combinations  of  figures  are  formed,  and  if 
the  same  result  is  obtained  by  both  operations  it  is  hardly  possible  that  it  can  be  wrong,  unless  the 
columns  are  not  evenly  spaced,  and  a figure  or  figures  are  added  in  with  the  wrong  column,  or  omitted. 

Addition  may  also  be  proved  by  “casting  out  the  9’s  ” in  each  line,  writing  the  remainder,  or 
“ excess  figure,”  to  the  right.  By  adding  these  several  remainders  and  casting  out  the  9’s,  or  by  car- 
rying the  remainder  from  each  line  into  the  next  line  and  continuing  back  and  forth  to  the  last  line, 
an  excess  figure  is  obtained  which  should  agree  with  that  obtained  by  casting  the  9’s  out  of  the  sum  of 
the  addition.  If  they  do  not  agree,  there  is  an  error  in  the  work. 

29.  In  addition  there  are  but  45  possible  combinations  of  two  figures;  the 
first  step  toward  proficiency  is  the  thorough  mastery  of  these  combinations,  so 
that  the  sum  of  any  two  figures  can  be  named  instantly. 


30.  The  combinations  of  two  figures,  one  above  another,  as  they  appear  to 

1 9 

the  eyes  in  adding  columns,  present  81  different  forms,  from  ^ to  g,  as  follows: 


1 

2 

1 

3 

1 4 

i 

5 1 

6 

l 

7 1 

8 

1 9 

2 

2 

3 

2 

2 

1 

3 

1 

4 1 

5 

1 6 

1 

7 

1 8 

1 

9 1 

2 

3 

2 

4 

4 

2' 

5 

2 

6 2 

7 

2 8 

2 

9 

3 3 

4 

3 5 

3 

6 

3 

/ 

2 

5 

2 

6 

2 7 

2 

8 2 

9 

2 

3 4 

3 

5 3 

6 

3 

i 

O 

O 

3 

8 

3 

9 

4 4 

5 

4 6 

4 

7 

4 8 

4 

9 5 

5 

6 

5 

i 

8 

3 

9 

3 

4 5 

4 

6 4 

7 

4 

8 4 

9 

4 5 

6 

5 

7 

5 

5 

8 

5 

9 

6 6 

7 

6 8 

6 

9 

7 7 

S 

7 9 

8 

8 

9 

9 

8 

5 

9 

5 

6 7 

6 

8 6 

9 

6 

7 8 

7 

9 7 

8 

9 

8 

9 

31. 

After  practising 

until  he  is 

; able  to  name  the 

) sum  of  any  of  th 

ese  < 

:om- 

bin 

ations  at 

a glance,  tin 

e student’s 

next  step 

is  to  lea 

rn 

to  read  : 

it  sight 

the 

sum 

of  f 

such 

combinations  as 

the  following : 

3 

r 

'7 

A 7 

7 q 

7 

7 

7 7 

,2  f 

7 

7 

A A 

a. 

aA 

71 

3- 

sSL 

. 

A. 

_xll 

A 

r 

7 

% 

7 A 

r f 

f 

r 

A3 

7 A 

SA 

7 

’A 

(o 

— 7- 

7_ 

A- 

7 

7? 

A 

7 

32.  After  a thorough  drill  on  the  two-figure  combinations  the  student  should 
begin  practise  upon  three-figure  groups.  The  following  are  a few  of  the  most 
practical  of  the  728  possible  combinations  of  three  figures.  It  will  be  found 
profitable  to  prepare  pages  of  similar  combinations,  and  to  practise  upon  them 
frequently  until  they  are  as  familiar  to  the  eye,  as  the  three-letter  words  one  sees 
in  books  and  papers.  Two  or  more  figures  should  not  be  regarded  as  separate 
characters,  but  as  forming  a word. 


ADDITION 


13 


THREE-FIGURE  COMBINATIONS  FOR  ‘ SIGHT  READING' 


2 

1 

3 

2 

5 

4 

7 

5 

2 

4 

4 

7 

6 

4 

2 

6 

5 

3 

1 

3 

4 

3 

4 

7 

4 

2 

1 

1 

6 

5 

3 

2 

1 

1 

3 

2 

2 

5 

7 

1 

4 

6 

3 

1 

1 

4 

2 

4 

2 

1 

3 

1 

3 

5 

5 

2 

3 

2 

2 

6 

3 

4 

2 

2 

2 

8 

3 

3 

7 

1 

5 

2 

6 

4 

1 

2 

3 

6 

Q 

O 

4 

2 

2 

7 

6 

5 

1 

7 

6 

2 

3 

4 

8 

2 

5 

2 

8 

2 

2 

4 

5 

6 

5 

2 

O 

O 

4 

2 

i 

2 

2 

7 

2 

1 

3 

2 

8 

2 

4 

5 

3 

5 

Q 

O 

2 

7 

6 

8 

2 

4 

3 

2 

G 

8 

3 

4 

9 

4 

3 

6 

< 

8 

2 

2 

6 

2 

1 

1 

9 

6 

6 

7 

5 

2 

7 

O 

O 

2 

2 

2 

4 

2 

2 

5 

7 

4 

3 

5 

3 

1 

2 

3 

3 

1 

2 

2 

5 

1 

7 

8 

3 

4 

3 

2 

6 

5 

2 

4 

2 

6 

4 

2 

7 

9 

1 

3 

5 

4 

3 

5 

8 

6 

7 

7 

3 

2 

8 

8 

9 

2 

3 

5 

4 

1 

8 

4 

6 

5 

7 

4 

4 

1 

5 

4 

4 

6 

O 

O 

1 

2 

5 

6 

6 

2 

3 

4 

6 

2 

4 

3 

4 

1 

6 

1 

4 

2 

4 

7 

3 

6 

9 

2 

5 

7 

3 

5 

4 

9 

2 

8 

5 

4 

5 

3 

1 

3 

2 

2 

4 

3 

2 

4 

2 

3 

8 

o 

O 

5 

4 

9 

2 

7 

O 

O 

6 

3 

9 

9 

8 

5 

l 

5 

3 

8 

7 

4 

3 

6 

5 

1 

3 

4 

2 

7 

3 

8 

o 

O 

7 

5 

8 

9 

4 

6 

5 

7 

9 

8 

3 

7 

9 

5 

8 

4 

6 

i 

6 

8 

9 

6 

7 

3 

7 

9 

7 

8 

4 

3 

9 

4 

5 

8 

9 

9 

5 

5 

7 

9 

4 

9 

5 

8 

9 

5 

8 

5 

7 9 

REVIEW 

8 

9 

6 

7 

3 

7 

9 

8 

7 

4 

CO 

\ 

8 

2 

3 

7 

9 

6 

5 

4 

5 

7 

6 

8 

7 

2 

9 

8 

4 

7 

6 

4 

3 

7 

6 

4 

Q 

O 

5 

7 

7 

4 

6 

2 

4 

2 

8 

4 

6 

5 

1 

5 

5 

4 

3 

4 

3 

4 

3 

4 

6 

5 

3 

5 

3 

5 

O 

O 

1 

2 

4 

6 

Remark. — Four,  five  and  six-figure  combinations  consist  simply  of  two  or  more  groups  of  two 
or  three  figures  each,  and  require  no  special  instruction  or  drill.  The  habit  of  recognizing  and  reading 
them  will  come  naturally  with  practise  after  proficiency  on  the  two  and  three-figure  combinations  has 
been  attained. 

33.  Addition  being  the  most  important  of  all  commercial  calculations, 
thorough  drill  in  its  rapid  performance  should  be  given  daily.  The  student  will 
get  much  practise  in  writing  up  accounts,  but  unless  this  is  supplemented  by 
daily  practise  for  speed,  he  will  not  become  rapid  and  accurate.  It  is  noticeable 
that,  as  a rule,  the  most  rapid  adders  are  the  most  accurate.  All  work  for  speed 
should  be  dictated,  and  the  time  required  for  the  operation  should  be  limited  or 
noted. 

The  subject  is  treated  for  beginners  and  advanced  students.  Discretion 
must  be  used  by  the  teacher. 


14 


ADDITION 


34.  A convenient  method  for  class-drill  is  to  give  tables,  as  shown  below,  of 
a prescribed  number  of  columns  and  rows,  suited  to  the  ability  of  the  students, 
the  figures  being  dictated  in  pairs  : 


S S £/  £ 2 

y y £ S3  33 

7 3 / 3 7 3 

2 / Sw  S3' 

32s/  y 2 

3 / y 2 3 2 

7 S 3 S 2 3 

S y 3 S2  S 

7 S 3 37  S 

3 2 y & i/  / 

2 / 2 y y 3 

S3 y S / S 

y 7 2 3 2 y 

S 2 3 2 3 y 

£ 2 S2  y £ 

2/72/2 

3 £ 2 .£  2 2 

2 3 3 S / S 

3 y 3 7 3 £ 

/ s 2 s / y 

7 2 2 3 S 3 

3 3 2 3 / 2 

. s s S3  y / 

3 3 Sy  £ 2 

/ y 3 z/  £ y 

23  y y 2 y 

23  £ y 2 7 

7 2 £ 3 S3, 

73  2 s £ s^ 

3 sy  / yjy 

3 7 3 £>3  3 

y y £ S y 3 

S3  3S 3 S 

3 y S 2 y S 

y / y £ 2 S 

2 2 3 / S y 

223 2 f 7 

y y y 3 £ y 

£ S £ 27  y 

S 7 y y 3 2 

3 7 S £ y s 

2 y y / 7 3 

y 2 *2  £ 3 / 

2 / 3 y 3 s 

7 y 3 y 3 / 

2 y y 2 3 y 

2 y y / S y 

7 2 £ 3 3 y 

s S y y y 3 

£22  £ 3 3 y 

3 3~  y £ £ 3 

y 3323  / 

y y y / y £ 

s'  33  y y 3 

3 3 y 3 2 s 

3 £ S 3 £ y 

y / s 2 2 3 

y £ S 3 7 s 

£7773  23 

£ 7 y s y z 

Note. — The  above  arrangement  is  recommended  for  sums  of  six  columns  in  ten  rows.  These 
are  known  as  “six  by  tens,”  and  should  be  dictated  as  follows  (taking  the  first  table  as  an  example)  : 
“Forty-five,  sixty-one,  sixty-two,  twenty-one,  forty-nine,  forty-five,’’  etc. 

For  beginners,  the  dictation  should  occupy  about  one  minute,  at  the  end  of  which  say,  “Add.’’ 
Allow  sixty  seconds  for  the  adding  of  a table,  and  begin  the  dictation  of  the  next  promptly,  continuing 
until  the  last  one  has  been  added,  wben  results  may  be  called  for  or  read  off  by  the  teacher  and 
checked  by  the  student.  At  first  very  few  students  will  be  able  to  obtain  the  correct  results  within 
the  time  limit,  but  as  they  become  more  expert  many  will  obtain  them  in  from  twenty  to  thirty 
seconds. 

In  adding  columns  insist  upon  the  student’s  naming  only  the  results  of  combinations,  omitting 
the  unnecessary  words  “and  are,”  or  “and  is.” 


ADDITION 


15 


35.  Tables  of  “ eight  by  fifteen,”  as  shown  below,  may  be  given  to  more 
advanced  students  in  the  same  manner  as  the  “ six  by  ten,”  having  the  time 
limit  of  ninety  seconds  or  less,  depending  on  the  accuracy  and  skill  of  students 
in  handling  figures.  Teachers  should  prepare  many  similar  examples  for  daily 
drill. 


57836489 

91445S27 

68563394 

26497764 

37195183 

25377836 

79213152 

57427435 

88239176 

98446712 

77492682 

3S768699 

82389724 

53668795 

26582358 

59344193 

47547362 

48993286 

62275846 

71323494 

94563324 

53974829 

72465137 

25784159 

84951873 

28959271 

29438196 

28926784 

43852971 

65267962 

35425723 

74884675 

54799321 

36616389 

64269762 

58517261 

89366339 

98523173 

22458165 

71951831 

83649177 

44277686 

55858487 

68665653 

84467125 

62785921 

79939831 

96439565 

29926997 

65823589 

49269482 

38243299 

24944828 

71S14554 

13234947 

34412847 

54714523 

32382649 

82992241 

49518736 

56339768 

53666987 

73935986 

76647862 

52679624 

92678429 

85439665 

44793198 

21252791 

63742615 

83764791 

44762786 

57885837 

36528939 

97643284 

36.  Addition  of  More  than  One  Column  in  One  Operation 


3 2. 
/ 3 
2 S' 
S / 

/ S S 


Process 

24  + 50=  74+1=  75 
75+20=  95+5=100 
100+10=110  + 3=113 
113  + 30=143+2=145 


This  method  proceeds  by  adding  the  tens  and  then  the  units. 

3 S 2 

/ Z / Process 

S y 2-  372+200+30+^+100+20+1+^0+40+2  = 1069. 

/ O 

The  student  should  be  encouraged  to  use  above  methods.  They  give  varia- 
tion of  method  and  train  to  quickness  in  mental  extension. 


16 


ADDITION 


37.  A common  practise  among  bookkeepers  and  bankers,  especially  when 
frequent  interruptions  occur,  is  to  indicate  at  the  right  the  sum  (including 
carrying  figure)  of  each  column,  the  result  being  the  addition  of  the  last  column 
and  the  last  figure  or  cipher  of  each  preceding  sum  in  regular  order.  By  the 
civil  service  method,  the  sums  of  the  columns  are  set  to  the  right  in  manner 
indicated  and  re-added. 


Bookkeepers’  Method 

Civil  Service  Method 

23  y b r y 

3 0 

3 o 

V b b cT  3 2 

*3  / 

j r 

y 4 i / & y 

3 S 

3 / 

2 3 b 7 44 

3 f 

3 b 

4 2 / /.  r / 

2 r 

2 S' 

<3  7 / S 2 

2 1/ 

2 2 

2 4/  f f S'/  0 

2 t/  f JS  / o 

CROSS  ADDITION  AND  TABULAR  WORK 

38.  The  ability  to  add  numbers  horizontally,  quickly  and  accurately,  is 
almost  as  valuable  as  the  ability  to  add  vertically.  Frequent  drills  should 
be  given,  and  care  should  be  taken  to  add  only  figures  of  the  same  order. 

WRITTEN  PROBLEMS 

39.  1.  Find  the  total  number  of  yards  in  ten  pieces  containing  the  follow- 
ing numbers  of  yards : 25,  42,  37,  29,  35,  51,  47,  39,  31,  43. 

2.  Cash  Sales 


Monday 

Tuesday  j 

Wednesday  1 

Thursday 

Friday 

Saturday  1 

Total 

$123.45 

$272.68  j 

$89.07 

$431.22 

$345.16 

$650.05 

3.  Coal  Shipments — Pounds 


Egg 

Stove 

Pea 

Buckwheat  Totals 

Monday 

327050 

472150 

72140 

27040 

Tuesday 

438160 

384060 

48120 

3S160 

Wednesday 

358100 

538010 

38100 

35100 

Thursday 

376080 

368070 

68170 

360S0 

Friday 

420120 

240180 

40080 

18040 

Saturday 

110060 

125100 

25010 

10020 

Totals 

Total  pounds  of  egg  coal  shipped  during  week  ? Stove  ? Pea  ? Buckwheat  ? 
Total  coal  shipped  each  day  ? Total  of  all  kinds  for  the  week  '? 


ADDITION 


17 


A Sums  disbursed  from  Peabody  Fund,  1876-1880,  inclusive. 


1876 

1877 

1878 

1879 

1880  Totals 

Virginia 

$17800 

$18250 

$15350 

$9850 

$6800 

North  Carolina 

8050 

4900 

4500 

6700 

3050 

South  Carolina 

4150 

4300 

3600 

4250 

2700 

Georgia 

3700 

4000 

6000 

6500 

5800 

Tennessee 

10100 

15850 

14600 

12000 

10900 

Totals 

5.  Five  salesmen  made  the  following  returns  of  sales  during  a week  : On 
Monday— No.  1,  $307.50;  No.  2,  $416.70;  No.  3,  $178.25;  No.  4,  $281.05;  No.  5, 
$198.67.  On  Tuesday — No.  1,  $209.10 ; No.  2,  $258.40;  No.  3,  $227.72;  No.  4, 
$203.50;  No.  5,  $268.80.  On  Wednesday— No.  1,  $201.48;  No.  2,  $165.10;  No. 
3,  $245;  No.  4,  $176.50;  No.  5,  $198.79.  On  Thursday— No.  1,  $146.60;  No.  2, 
$160.36;  No.  3,  $236.45;  No.  4,  $245.15;  No.  5,  $35L70.  On  Friday— No.  1, 
$185.50;  No.  2,  $240.30;  No.  3,  $149.30;  No.  4,  $194.82;  No.  5,  $282.70.  On 
Saturday— No.  1,  $301.70;  No.  2,  $337.40;  No.  3,  $341.80;  No.  4,  $380.78;  No.  5, 
$417.80.  Arrange  in  tabular  form  and  answer  the  following  questions:  Lpon 

what  day  was  the  greatest  amount  of  sales  made?  Which  salesman  sold  the 
greatest  amount  of  goods?  What  was  the  total  amount  of  sales  for  the  week? 


The  following  are  the  daily  receipts  of  the  treasurer  of  a borough  for 
fourteen  consecutive  weeks  : 


Weeks 

Monday 

Tuesday 

Wednesday 

Thursday 

Friday 

Saturday  Totals 

First 

486 

24 

360 

50 

48 

21 

516 

18 

233 

40 

10s 

96 

Second 

132 

25 

897 

56 

437 

52 

213 

78 

478 

09 

980 

43 

Third 

731 

64 

930 

4S 

376 

83 

481 

63 

704 

45 

173 

26 

F ourtli 

897 

26 

270 

45 

103 

48 

862 

24 

238 

27 

527 

28 

Fifth 

188 

25 

390 

31 

976 

82 

932 

21 

752 

34 

438 

21 

Sixth 

297 

11 

434 

56 

542 

26 

276 

43 

177 

30 

222 

43 

Seventh 

729 

55 

721 

14 

297 

31 

274 

90 

249 

32 

576 

87 

Eighth 

253 

54 

490 

78 

864!44 

831 

82 

757 

42 

543 

85 

Ninth 

238 

80 

809 

66 

198 

23 

582 

64 

523 

45 

350 

67 

Tenth 

487 

56 

789 

40 

169 

45 

346 

24 

434 

56 

456 

78 

Eleventh 

913 

27 

841 

54 

786 

32 

123 

45 

876 

43 

784 

27 

Twelfth 

1963 

29 

1187 

95 

864 

21 

487 

91 

798 

63 

1428 

95 

Thirteenth 

1048 

63 

968 

42 

1298 

74 

689 

40 

1698 

75 

1078 

64 

Fourteenth 

857 

93 

1236 

97 

1187 

42 

1693 

75 

1574 

83 

1870 

71 

Totals 

--s> 

1 

V 

r> 

■1\S\ 

Find  the  total  receipts  for  each  day ; total  receipts  for  each  week  ; total 
receipts  for  the  fourteen  weeks. 


SUBTRACTION 


40.  Subtraction  is  the  process  of  finding  the  difference  between  two  numbers. 

41.  The  Difference  between  two  numbers  is  a number  which,  added  to  the 
less,  will  make  it  equal  to  the  greater. 

42.  The  Subtrahend  is  the  number  to  be  subtracted. 


43.  The  Minuend  is  the  number  from  which  the  subtrahend  is  taken. 

44.  The  Sign  of  Subtraction  is — , read  minus,  and  denotes  that  the  num- 
ber following  it  is  to  betaken  from  the  number  preceding  it. 

45.  The  Principles  of  Subtraction  are : 

1.  Only  like  numbers  can  be  subtracted. 

2.  Units  of  the  same  order  only  can  be  directly  subtracted. 

3.  The  difference  is  a number  of  the  same  kind  as  the  minuend  and  sub- 
trahend. 

f.  If  the  minuend  and  subtrahend  be  equally  increased  or  diminished,  the 
difference  remains  the  same. 


46.  To  find  the  difference  between  two  numbers. 


Examples. — (1)  Subtract  476  from  847 ; (2)  from  10000  take  1346. 


^ After  writing  the  subtrahend  under  the  minuend  so  that 

/ units  are  under  units,  tens  under  tens,  etc.,  we  begin  with 

— ‘T-  7 fio  units  and  say,  “Six  from  seven  leaves  one,”  and  write  1 / 3 *-/■  4 

% y ^ under  the  units  column.  In  the  tens  column  we  cannot  ^ 

' take  7 from  4,  so  we  borrow  1 (hundred)  from  the  8,  which 

equals  10  tens,  and  add  it  to  the  4 and  say,  “ Seven  from  fourteen  leaves  seven,’’  and  write  7 under  the 
tens.  Having  decreased  8 by  borrowing  from  it  we  now  say,  “Four  from  seven  leaves  three,”  and 
write  3 under  the  hundreds,  which  gives  for  a result,  371 — the  difference. 


Note. — Another  method  of  subtracting  which  has  its  advantages  in  such  cases  as  Example  2.  is 
as  follows  : Instead  of  borrowing  1 from  the  highest  order  and  regarding  it  as  a succession  of  9’s  in 
each  of  the  other  orders  excepting  the  units,  where  it  must  be  considered  as  10,  the  operation  is  per- 
formed by  saying,  “Six  from  ten  leaves  four ; five  from  ten  leaves  five  ; four  from  ten  leaves  six  ; two 
from  ten  leaves  eight” — increasing  the  subtrahend  figure  by  1 at  each  step  after  the  first,  and  regard- 
ing the  minuend  each  time  as  10. 

47.  R ule. — Write  the  subtrahend  under  the  minuend  so  that  units  of  the  same 
order  stand  in  the  same  column. 

Beginning  with  the  units,  subtract  each  figure  from  the  one  above  it,  uniting  the 
difference  under  the  figure  subtracted. 

If  any  figure  in  the  minuend  is  less  than  the  figure  to  be  subtracted,  increase  it  by 
ten,  by  taking  1 from  the  next  order  above,  then  subtract. 


18 


SUBTRACTION 


19 


48.  Proof. — Add  the  difference  to  the  subtrahend  ; if  correct,  the  sum  will 
equal  the  minuend. 

Another  proof  of  subtraction  is  to  cast  out  the  9’s  in  the  minuend  and  sub- 
trahend, then  subtract  the  excess  figure  of  subtrahend  from  excess  figure  of  min- 
uend. If  the  latter  is  the  smaller,  add  9.  The  difference  must  equal  excess 
figure  of  the  remainder. 

49.  The  following  exercises  are  merely  suggestive.  When  a figure  in  the 
minuend  is  the  smaller,  borrow  a unit  from  the  next  higher  order  which  is  equal 
to  ten  of  th.e  lower  order.  Add  ten  and  then  subtract. 

In  oral  drills,  avoid  such  expressions  as  “ from  ” and  “ leaves.'’  Name  results 
only. 

MENTAL  EXERCISE 


7 

9 8 

6 

5 7 

8 

6 

9 

i J 

8 

6 

8 

9 

3 

4 5 

2 

3 4 

2 

4 

3 

5 2 

o 

O 

3 

6 

6 

27 

36 

45 

18 

19 

37 

39 

35 

28 

38 

48 

12 

14 

13 

13 

12 

11 

15 

14 

17 

18 

23 

49 

56 

45 

48 

54 

47 

65 

44 

36 

37 

67 

33 

■ 32 

21 

27 

30 

35 

43 

23 

23 

22 

25 

57 

76 

84 

96 

79 

84 

59 

46 

86 

75 

66 

23 

32 

31 

43 

67 

52 

24 

21 

53 

21 

22 

43 

32 

34 

37 

51 

34 

41 

36 

50 

42 

35 

19 

25 

18 

19 

16 

15 

19 

17 

21 

27 

19 

46 

53 

49 

40 

52 

34 

47 

64 

45 

52 

32 

18 

26 

23 

22 

23 

ORAL 

25  19 

EXERCISE 

27 

28 

23 

19 

50.  1-  From  28  take  17. 
take  19. 

From  63  take  36. 

From  52  lake 

25. 

From 

35 

5 from  9 leaves  what? 
from  56  ? 34  from  43  ? 

3 from  8 ? 
( 

15 

from  35? 

37 

from  73? 

IS 

3.  28 — 19= 

= ? 34 

-23=  ? 

37 — 

28=  9 

75 — 59=  ? 37 

—18= 

_ 9 

If..  What  is  the  difference  between  34  and  57?  48  and  29?  85  and  57? 
91  and  19?  57  and  75?  49  and  94? 

5.  What  is  the  difference  between  $17  and  $65?  $83  and  $29?  $67  and 

$83  ? $.53  and  $.27  ? $.67  and  $1  ? $.75  and  $2  ? 

6.  Two  boys  had  respectively  $1.10  and  $.85;  if  each  spends  $.35,  how 
much  money  will  they  have  left? 

7.  A boy  had  73  cents  and  his  father  gave  him  a half  dollar ; if  he  spends 
48  cents,  how  much  has  he  remaining? 


20 


SUBTRACTION 


8.  Two  newsboys  together  bought  newspapers  to  the  value  of  75  cents;  one 
sold  55  cents  worth,  the  other  65  cents  worth.  What  did  the\r  gain? 

9.  A man  has  two  fields,  one  of  32  acres,  the  other  of  57  acres;  he  puts  48 
acres  in  wheat  and  the  rest  in  corn;  how  many  acres  in  corn? 

10.  A man  sold  two  horses  for  $135  and  $110,  respectively;  what  did  lie 
gain  or  lose,  if  he  had  paid  $250  for  the  pair? 

11.  What  is  the  value  of  13+15—9?  Of  23  + 33—17?  Of  83—50+23? 
Of  24+48— 13?  Of  55— 16— 7 + 6? 

IS.  Add  46  to  33  and  take  away  27.  Take  away  29  from  77  and  add  15. 

13.  Add  7,  9,  8,  6 and  5 and  from  the  result  take  the  sum  of  6,  4,  7. 

7+  I sell  a piano  for  $225,  and  receive  cash  $165  and  a note  to  balance 
account.  What  is  the  face  of  the  note? 

15.  A merchant  buys  a bill  of  merchandise  for  $383,  giving  a note  for  $175 
and  cash  for  balance.  How  much  cash  does  he  pay? 


CASHIERS’  CALCULATIONS 


51.  1.  A salesman  handed  in  a sales  check  for  $3.73  and  a $5  bill.  Hand 
him  the  change.  Ans. — 2 pennies,  a quarter  and  a dollar. 

S.  Required  the  change,  in  the  fewest  denominations,  for  a sale  of  $4.19  to 
be  taken  out  of  a $5  bill.  Ans. — 1 penny,  1 nickel,  1 quarter,  1 half. 


3.  Amount  of  sale. 

Money  tendered. 

$ 3.19 

$10  bill 

.75 

5 bill 

3.79 

20  bill 

.18 

2 bill 

1.17 

1 bill  and  $J  silver 

15.68 

Two  10  bills 

4.49 

10  bill 

3.33 

5 bill 

8.90 

5 bill  and  two  $2  bills 

.65 

2 bill 

14.91 

20  bill 

1.93 

5 bill 

1.67 

1 bill  and  $4  and  SJ  silver 

13.23 

10  bill  and  $5  bill 

3.09 

2 bill  and  $1  and  $^  silver 

11.21 

50  bill 

83.37 

100  bill 

68.12 

Four  20  bills 

16.71 

10  bill  and  $5  bill  and  82  bill 

SUBTRACTION 


21 


f A salesman’s  check  shows  items  35  cents,  70  cents  and  18  cents;  a $20 
bill  is  tendered.  What  change  is  handed  back  ? 

5.  Salesman’s  items.  Money  tendered. 

$.50,  .20,  .18  $2  bill 

.39,  .15,  .25,  .10  5 bill 

.75,  .20,  .30  10  bill 

.36,  .13,  1.20  2 bill 

.87,  .33,  .37  20  bill 

52.  A good  mental  drill  in  subtraction  is  to  prepare  lists  of  subtrahends  to 
be  taken  from  50,  100,  500,  1000,  etc.  Start  with  100  as  a fixed  minuend,  and 
call  off  in  rapid  succession  all  subtrahends  from  10  to  20,  in  irregular  order;  as 
“ fifteen,”  “ twelve,”  “ eighteen,”  etc.,  and  have  students  write  down  the  remainders  ; 
as  85,  88,  82,  etc.  Finish  with  20,  and  then  call  off  in  irregular  order  the  num- 
bers between  20  and  30,  finishing  with  30.  Proceed  in  this  manner  by  increasing 
subtrahends,  the  students  putting  down  the  decreasing  remainders.  From  time 
to  time  call  off  the  remainders  and  have  students  check  their  results. 

In  using  50  for  a fixed  minuend,  begin  with  25  as  your  subtrahend  and 
work  alternately  up  and  down  from  this  point.  After  some  proficiency  has  been 
attained  in  these,  500  or  1000  may  be  used  for  the  minuend  in  the  same  manner. 
This  exercise  may  be  used  for  oral  drill  also ; but  for  class  drill  the  written  work 
has  the  advantage  of  being  quieter,  and  also  of  benefiting  in  a greater  degree  the 
slower  students  who  are  generally  disposed  to  remain  silent  in  concert  oral  drills. 


WRITTEN  PROBLEMS 
Balancing  Accounts 

Example. — Find  the  balance  of  the  following  account : 
Dr.  Cr. 


2 3 7-7^ 

r / 2 O f 

V 3 ■ 6 7 

r 2 . f 6 

/ 3 3.BB 


/ 3 0 


f'f  * 


r 17  o 0.0  <2 
600.00 
r o . 0 0 

Bala  nee,  2.  A B . f?  2. 

" / 3 0 f . f A 


53.  Rule. — Find  the  sum  of  the  larger  side , set  the  total  beneath  : add  the  columns 
of  the  other  side,  setting  down  in  the  place  for  the  balance  the  figure  necessary  to  make 
the  required  figure  in  the  total : verify  by  adding  and  totaling  all  the  items  of  the 
smaller  side. 

54.  1.  Find  the  balance  of  the  following  account: 

Debits  $243.75;  $18.94;  $309.90;  $475.60. 

Credits  385.20;  10.50;  930.27;  50.40. 

Note. — Arrange  amounts  in  columns  and  find  balance  as  in  model. 


22 


SUBTRACTION 


2.  The  debit  items  of  a trial  balance  are  $644.50,  $454.70,  $27,  $44.80.  $104; 
and  the  credit  items,  excepting  capital  account,  are  $212.50,  $62.50.  What  is  the 
amount  of  capital  ? 

3.  Cash  items  received  during  a day  were  $1000,  $28.50,  $63,  $75  ; and  the 
cash  items  paid  out  were  $300,  $50,  $100,  $72.  What  is  the  balance  on  hand? 

J.  The  amount  of  bills  receivable  received  during  a month  were  $400,  $100, 
$200,  $150,  and  the  amount  paid  on  them  was  $150.  How  much  remains  unpaid  ? 

5.  Merchandise  items  bought  in  a month  were  respectively:  $2440,  $74.50, 
and  $901.25.  Sales  made  $286.25,  $400,  $116.25,  $314,  $441.50,  $572,  $240,  $309, 
$227.50,  and  $755.25.  If  all  sold,  what  amount  of  gain  or  loss  does  the  balance 
show  ? 

6.  The  debit  items  of  a loss  and  gain  account  are  as  follows:  $250,  $225, 

$153.60,  $150,  $254.76,  $398.83,  $12,  $8.17,  $71.27,  $17.50.  The  credit  items  are 
$14.53,  $150,  $310.  What  is  the  balance  of  the  account,  and  does  it  show  a gain 
or  a loss  ? 

7.  During  the  month  of  March  a merchant  allowed  discounts  to  his 

customers  as  follows  : $2.20,  $5,  $1.25,  $1.53,  $1.83,  $14.58.  He  was  allowed  dis- 

counts by  others  $1.75,  $25.39,  $4.20,  $9.58.  Does  his  discount  account  show  a 
gain  or  loss,  and  how  much? 

8.  A merchant  deposited  in  bank  during  a month  $8692.60,  $11833.93, 
$7933.93,  and  checked  out  $1600,  $2800,  $2670.60,  $5870.60,  $5792.60.  What  is 
his  balance  in  bank  ? 

9.  A farmer  made  an  inventory  of  his  effects  and  found  that  he  had  land 
worth  $12600,  buildings  $3450,  live  stock  $2365,  wheat  $218.75.  farming  imple- 
ments $680,  household  furniture  $1475,  J.  Wright’s  note  $300.  He  owed  a mort- 
gage of  $6000,  notes  held  by  others  $1575,  and  outstanding  accounts  $2367.80. 
What  was  his  net  worth  ? 

10.  A business  has  cash  on  hand  $11780,  notes  on  hand  $650.  It  owes 
$3475.  What  is  the  net  worth  of  the  business  ? 

11.  A business  man  makes  the  following  bank  deposits  : S1578.25.  $614. 
$342.50,  $595.69,  $734,  $110.10  ; and  draws  the  following  checks : $224.73,  $219.20, 
$163.57,  $163.75,  $591.32,  $176.67.  What  is  his  balance  in  bank? 

12.  Gash  balance  on  hand,  $9462.50 ; receipts  for  the  day,  $14165.75:  cash 
paid  out,  $18328.62.  Find  cash  balance  at  close  of  day. 

13.  To  the  merchandise  credits,  $17342.33,  add  the  value  of  inventory, 
$4555.70,  and  take  away  the  merchandise  debits,  $21876.66.  What  was  gained? 

11/..  Jones’s  account  shows  sundry  debits  as  follows:  $345.50,  $427.15.  $S7.93. 
$1503.34,  and  the  following  credits  : $250,  $300.50,  $475,  $290.90,  $75.8100.  $175, 
$120.25.  What  is  the  balance  of  his  account  ? 

15.  The  gross  weight  of  a car  and  its  contents  is  82525  pounds;  if  the  car 
weighs  38500  pounds,  what  is  the  weight  of  the  contents  ? 

16.  The  resources  of  a business  are:  Cash  $2351.23,  Merchandise  357S1.16. 

Real  Estate  $8200,  Bills  Receivable  $1250,  Personal  Accounts  due  81462.50.  The 
liabilities  are:  Bills  Payable  $2550,  Personal  Accounts  owing  $714.69.  The 
capital  is  the  excess  of  the  resources  above  the  liabilities.  What  is  its  amount  ? 


SUBTRACTION 

23 

17.  An  erroneous  trial  balance  is 

as  follows: 

Dr. 

Cr. 

T.  W.  Baylor 

$1600.00 

$15000.00 

Cash 

6C000.00 

55000.00 

Real  Estate 

4000.00 

5000  00 

Merchandise 

21000.00 

27500.00 

Expense 

4500.50 

Bank  Stock 

800.00 

1050.50 

R.  R.  Stock 

1000.00 

Interest 

350.75 

100.75 

Commission 

95.00 

Personal  Accounts 

13910.00 

Personal  Accounts 

2499.00 

How  much  is  it  out  ? 

18.  Check  Stub.  Add  deposits. 

Subtract  checks  drawn. 

Balance 

$1500.00 

Deposits 

465.50 

Drawings 

332.25 

Balance 

Deposits 

825.10 

Drawings 

342.90 

Balance 

Deposits 

119.62 

Drawings 

485.00 

Balance 

Deposits 

526.30 

Drawings 

310.80 

Balance 

Deposits 

48.60 

Drawings 

1200.00 

Balance 

19.  Arrange  items  and  perform  operation  as  in  problem  No.  18. 

William  Bray  had,  on  Monday,  a bank  balance  of  $685.61 ; he  deposited 
$412.80  and  drew  checks  for  $122.59,  $145.80,  and  $157.10.  On  Tuesday  he  drew 
checks  for  $158.99,  $100.67,  $149.27,  and  $120.  What  is  the  balance  to  his  credit 
on  Tuesday  night  ? 

SO.  John  Cake  had,  on  Monday,  a bank  balance  of  $22.60  and  deposited 
$768.25.  On  Tuesday  he  drew  checks  for  $156.28,  $191.75,  $156.28,  and  $201. 
What  is  the  balance  to  his  credit  on  Tuesday  night  ? 


MULTIPLICATION 


55.  Multiplication  is  the  process  of  finding  the  product  of  two  numbers. 

56.  The  Product  of  two  numbers  is  the  sum  obtained  by  adding  one  of  the 
numbers  to  itself  as  many  times  as  there  are  units  in  the  other  number. 

57.  The  Multiplicand  is  the  number  to  be  multiplied. 

58.  The  Multiplier  is  the  number  which  shows  how  many  times  the  multi- 
plicand is  to  be  repeated. 

59.  The  Sign  of  Multiplication  is  X,  read  multiplied  by,  and  denotes  that 
the  number  preceding  it  is  to  be  multiplied  by  the  number  following  it. 

60.  The  Principles  of  Multiplication  are: 

1.  The  product  is  a number  of  the  same  kind  as  the  multiplicand. 

2.  The  multiplier  is  art  abstract  number,  because  it  denotes  the  number  of 
times  the  multiplicand  is  repeated  to  produce  the  product. 

3.  The  product  of  two  abstract  numbers  is  the  same,  whichever  is  made  the 
multiplier. 

Ip.  If  the  multiplier  be  separated  into  parts,  and  the  multiplicand  be  multi- 
plied by  each  part  separately,  the  sum  of  the  partial  products  will  be  the  entire 
product ; as,  fifteen  times  a number  is  equal  to  ten  times  the  number  plus  five 
times  the  number. 


61.  To  find  the  product  of  two  numbers. 


Example. — Multiply  436  by  2S7. 


Write  the  multiplier  beneath  the  multiplicand,  and  dravXa  line.  Seven 
times  6 units  are  42  units — write  2 directly  under  the  7 (so  that  it  will  be  in 
units  column ) and  carry  4 tens.  Seven  times  three  tens  are  21  tens  (4  tens- 
carried)  are  25  tens — write  5 in  the  tens  column  and  carry  2 hundreds. 
Seven  times  4 hundreds  (2  hundreds  carried)  are  30  hundreds — write  30. 

Eight  times  6 are  48 — write  8 under  the  tens  figure  (5)  of  the  first  par- 
tial product,  and  carry  4.  Eight  times  3 (4  carried)  are  28  hundreds — write 
8 under  hundreds  column,  and  carry  2.  Eight  times  4 (2  carried)  are  34 — 
write  34. 

Two  times  6 are  12— write  2 under  hundreds  column,  and  carry  1.  Two  times  3 (1  carried)  are 
7 — -write  7 under  thousands  column.  Two  times  4 are  8 — write  8 under  ten-thousands  column. 


U 3 

4 f 7 

3 0 S3 
3 r 

f 72 

/ 2- / 3 2l 


Draw  a line  and  add  the  three  partial  products,  which  gives  125132,  or  287  times  436. 


24 


MULTIPLICATION 


25 


62.  Rule. — Write  the  multiplier  under  the  multiplicand , so  that  units  of  the 
same  order  shall  stand  in  the  same  column. 

Beginning  with  the  units,  multiply  the  multiplicand  by  each  figure  of  the  multi- 
plier, separately,  writing  the  first  figure  of  each  partial  product  under  that  figure  of  the 
multiplier  which  produced  it 

Add  the  partial  products. 

63.  Proof. — Multiply  the  multiplier  by  the  multiplicand ; if  the  same 
result  is  obtained,  it  is  almost  sure  to  be  correct. 

A shorter  method  is  to  cast  out  the  9’s  of  the  multiplicand  apd  multiplier 
and  multiply  the  excess  figures  together.  If  correct,  the  excess  figures  of  the 
two  products  will  agree. 

64.  The  multiplication  of  any  two  numbers,  however  large,  is  but  a succes- 
sion of  multiplications  of  a single  figure  by  a single  figure.  Any  multiplication 
whatever  may  be  performed  by  the  student  who  has  mastered  the  followingjtable, 
which  contains  all  the  products  of  any  two  of  the  figures  2,  3,  4,  5,  6,  7,  8,  9 : 


MULTIPLICATION  TABLE 


2 

X 

2 = 

= 4 

O 

O 

X 

3 = 

= 9 

4 

X 

4 = 

= 16 

5 

X 

5 = 

= 25 

2 

X 

3 = 

= 6 

3 

X 

4 = 

= 12 

4 

X 

5 = 

= 20 

5 

X 

6 = 

= 30 

2 

X 

4 = 

= 8 

3 

X 

5 = 

= 15 

4 

X 

6 = 

= 24 

5 

X 

7 = 

= 35 

2 

X 

5 = 

= 10 

3 

X 

6 = 

= 18 

4 

X 

7 = 

= 28 

5 

X 

8 = 

= 40 

2 

X 

6 = 

= 12 

3 

X 

7 = 

= 21 

4 

X 

8 = 

= 32 

5 

X 

9 = 

= 45 

2 

X 

7 = 

= 14 

O 

O 

X 

8 = 

= 24 

4 

X 

9 = 

= 36 

6 

X 

6 = 

= 36 

2 

X 

8 = 

= 16 

3 

X 

9 = 

= 27 

7 

X 

7 = 

= 49 

6 

X 

7 = 

= 42 

2 

X 

9 = 

= 18 

8 

X 

8 = 

= 64 

7 

X 

8 = 

= 56 

6 

X 

8 = 

= 48 

9 

X 

9 = 

= 81 

8 

X 

9 = 

= 72 

7 

X 

9 = 

= 63 

6 

X 

9 = 

= 54 

65.  While  the  foregoing  table  is  all  that  is  absolutely  necessary,  the  student, 
in  order  to  become  a rapid  calculator,  should,  from  time  to  time,  increase  his  store 
of  memorized  products  by  careful  study  and  drill  upon  the  following  extended 
tables,  until  he  can  name  instantly  the  product  of  any  two  numbers  up  to  25 
times  25.  This  will  enable  him  to  perform  with  great  rapidity  many  commercial 
calculations,  especially  in  making  extensions  on  invoices,  etc. 


26 


MULTIPLICATION 


Multiplication  Table  Extended  to  12X12 


1 

2 

9 

O 

4 

5 

6 

7 

8 

9 

10 

11 

12 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

66 

72 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

132 

12 

24 

36 

48 

60 

72 

84 

96 

10S 

120 

132 

144 

Note. — The  numbers  in  bold-faced  type — products  of  two  equal  factors,  or  a number  multi- 
plied by  itself — are  called  squares ; as,  16  is  the  square  of  4. 


ORAL  MULTIPLICATION  DRILL 

66.  A very  useful  drill  to  enable  the  student  to  master  readily  the  extended 
tables  is  as  follows:  Name  all  the  products  of  one  factor  in  regular  order  from 
the  lowest  to  the  highest,  by  mentally  adding  at  each  step,  as  in  addition,  and 
then  repeat  them  again  from  highest  to  lowest,  mentally  subtracting  at  each  step, 
as  in  subtraction. 

It  is,  of  course,  necessary  to  name  the  factors  as  well  as  the  products — other- 
wise it  would  be  merely  an  addition  and  subtraction  drill.  “Twice  13  are 
26,  three  times  13  are  39,  four  times  13  are  52,  five  times  13  are  65,  six  times  13 
are  78,”  etc.,  to  “ twenty-five  times  13  are  325.”  Then  reverse — “Twenty-four 
times  13  are  312,  twenty-three  times  13  are  299,”  etc. 


MULTIPLICATION 


27 


Multiplication  Table  Extended  to  20  X 20 


1 

2 

3 

4 

5 

6 

7 8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

13 

26 

39 

52 

65 

78 

91 

104 

117 

130 

143 

156 

169 

182 

195 

208 

221 

234 

247 

260 

14 

28 

42 

56 

70 

84 

98 

112 

126 

140 

154 

168 

182 

196 

. 210 

224 

238 

252 

266 

280 

15 

30 

45 

60 

75 

90 

105 

120 

135 

150 

165 

180 

195 

210 

225 

240 

255 

270 

285 

300 

16 

32 

48 

64 

80 

96 

112 

128 

144 

160 

176 

192 

208 

224 

240 

256 

272 

288 

304 

320 

17 

34 

51 

68 

85 

102 

119 

136 

153 

170 

187 

204 

221 

238 

255 

272 

289 

306 

323 

340 

18 

36 

54 

72 

90 

108 

126 

144 

162 

18  ' 

198 

216 

234 

252 

270 

288 

306 

324 

342 

360 

19 

38 

57 

76 

95 

114 

133 

152 

171 

190 

209 

228 

247 

266 

285 

304 

323 

342 

361 

380 

20 

40 

60 

80 

100 

120 

140 

160 

180  200 

220 

240 

260 

280 

300 

320 

340 

360 

380 

400 

Multiplication  Table  Extended  to  25  X 25 


l 

o 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

21 

42 

63 

84 

105 

126 

147 

168 

189 

210 

231 

252 

273 

294 

315 

336 

357 

378 

399 

420 

441 

462 

483 

504 

525 

22 

44 

66 

88 

110 

132 

154 

176 

198 

220 

242 

264 

286 

308 

330 

352 

374 

396 

418 

440 

462 

484 

506 

528 

550 

23 

46 

69 

92 

115 

138 

161 

184 

207 

230 

253 

276 

299 

322 

345 

368 

391 

414 

437 

460 

483 

506 

529 

552 

575 

24 

48 

72 

96 

120 

144 

168 

192 

216 

240 

264 

288 

312 

336 

360 

384 

408 

432 

456 

480 

504 

528 

552 

576 

600 

25 

50 

75 

100 

125 

150 

175 

200 

225 

250 

275 

300 

325 

350 

375 

400 

425 

450 

475 

500 

525 

550 

575 

600 

| 

625 

1 

ORAL  EXERCISE 

67.  Ex.  —If  a train  runs  15  miles  an  hour,  how  far  will  it  run  in  8 hours? 
Ans. — It  will  run  8 times  15  miles,  which  is  120  miles. 

68.  1.  W hat  will  15  roses  cost  at  5 cents  each? 

2.  At  9 cents  a yard,  what  will  18  yards  of  ribbon  cost? 

3.  A clerk  earns  $14  a wreek ; how  much  will  he  earn  in  9 weeks? 

A steamer  goes  14  miles  an  hour;  how  many  miles  will  it  go  in  11 

hours  ? 

5.  What  will  be  the  cost  of  6 oranges  at  5 cents  each  and  12  lemons  at  3 
cents  each  ? 

6.  What  is  the  value  of  7 cows  at  $25  each  ? 

7.  What  will  be  the  cost  of  7 yards  muslin  at  12  cents  a yard,  3 spools 
thread  at  2 cents  each,  and  a paper  of  needles  5 cents? 


28 


MULTIPLICATION 


8.  If  three  men  can  do  a piece  of  work  in  14  days,  how  long  would  it  require 
one  man  to  do  it? 

9.  B having  35  sheep  sold  18  of  them  and  then  bought  3 times  as  many  as 
he  had  left ; how  many  did  he  then  have  ? 

10.  An  engine  travels  30  miles  an  hour  and  a trolley  car  22  miles  an  hour: 
how  much  farther  will  the  engine  travel  in  8 hours  than  the  trolley  car? 


SALESMEN’S  AND  CASHIERS’  CALCULATIONS 


69.  Required  the  amount  of  change  in  the  fewest  denominations  for  each 
of  the  following  sales  : 


Items  of  Sale 

Money  Tendered 

1. 

6 lbs.  Coffee 

@ 

$.38 

$2  bill 

2. 

3 “ Tea 

U 

.60 

Two  silver  dollars 

3. 

25  “ Sugar 

(6 

.05 

$1  bill  and  $4  silver 

1 

24  cans  Corn 

a 

.06 

5 bill 

5. 

6 doz.  Oranges 

u 

.30 

2 bill 

6. 

12  cans  Tomatoes 

u 

.08 

2 bill 

7. 

5 lbs.  Tea 

u 

.75 

10  bill 

8. 

11  “ Beef 

u 

.12 

5 bill 

9. 

6 “ Butter 

u 

.27 

2 bill 

10. 

8 “ Prunes 

u 

.14 

5 bill 

11. 

10  “ Soap 

(< 

.07 

2 bill 

12. 

5 “ Starch 

.08 

1 bill 

13. 

6 “ Rice 

u 

.07 

4 silver 

u: 

3 yds.  Silk 

u 

.65 

5 bill 

15. 

12  “ Ribbon 

a 

.15 

5 bill 

16. 

12  spools  Cotton 

u 

.03 

2 bill 

17. 

12  yds.  Chintz 

il 

.18 

5 bill 

18. 

25  “ Muslin 

u 

.08 

10  bill 

19. 

6 pairs  Hose 

u 

.25 

5 bill 

20. 

12  yds.  Flannel 

a 

.18 

2 bill  and  $1  silver 

WRITTEN  EXERCISE 


70.  1.  Multiply  $475.25  by  25. 

2.  Multiply  28375  by  153. 

3.  Multiply  62137  by  328. 
Multiply  1756  by  488. 

5.  Multiply  3991  by  998. 

6.  Multiply  $486.65  by  117. 

7.  Multiply  3937  by  4256. 

8.  Multiply  5236  by  973. 

9.  Multiply  215042  by  9364, 

10.  Multiply  231  by  9997. 


11.  Multiply  28317  by  472. 

12.  Multiply  6452  by  324. 

13.  Multiply  28375  by  756. 
U.  Multiply  25293  by  4938. 

15.  Multiply  311035  by  426. 

16.  Multiply  16387  by  729. 

17.  Multiply  $442.37  by  674. 

18.  Multiply  5029  by  943. 

19.  Multiply  $316.75  by  749. 

20.  Multiply  91442  by  3664. 


SHORT  METHODS 


29 


SPECIAL  RULES  AND  SHORT  METHODS 


71.  To  multiply  by  10, 100, 1000,  etc. 

72.  Rule. — At  the  right  of  the  multiplicand  place  as  many  ciphers  as  are  found 
in  the  multiplier. 

73.  To  find  the  product  of  two  numbers  when  either  or  both  end  in 
ciphers. 


Example. — Multiply  2590  by  38000. 

74.  Rule. — Find  the  product  of  the  significant 
figures  of  the  two  factors,  and  to  the  right  annex  as  many 
ciphers  as  are  found  at  the  right  of  both. 


i y / 
3 ri 

0 

O O O 

2 0/2 
7 7 7 

f M ^ 2- 

O O O 0 

Note. — The  judgment  of  the  teacher  must  he  exercised  as  to  the  introduction  of  short  rules. 
Some  may  be  used  to  advantage  with  beginning  grades  ; others  with  more  advanced  students. 

75.  To  multiply  by  25. 


Example.— Multiply  31416  by  25. 

„ . By  annexing  two  ciphers  the  number  is  multiplied  by  100, 

7 v m t-A  OO  or  4 times  the  value  of  25. 


I/)  3 / O O 


76.  Rule. — Multiply  by  100  (by  annexing  2 ciphers ) and  divide  by  f 


EXERCISE 


77. 


1.  Multiply  3048  by  25.  6. 

2.  Multiply  9144  by  25.  7. 

3.  Multiply  5029  by  25.  8. 

f Multiply  16093  by  25.  9. 

5.  Multiply  6452  by  25.  10. 


Multiply  92903  by  25. 
Multiply  8361  by  25. 
Multiply  25293  by  25. 
Multiply  4047  by  25. 
Multiply  16387  by  25. 


78. 


To  multiply  by  250,  125,  etc.,  is  but  an  extension  of  the  above  rule. 
Multiply  by  1000  (by  annexing  three  ciphers)  and  divide  by  4 and  8 respec- 
tively, 1000  being  4 times  the  value  of  250,  or  8 times  the  value  of  125.  Use 
Exercise  77  for  drill. 


79.  To  multiply  by  a number  a little  less  or  more  than  100, 1000,  etc. 

Example. — Multiply  3937  by  998. 


3/3/000 

f f'  Y i/-  (3937x2)  Subtract. 

3 / 2-  7 / 2-  3 

80.  Rule. — Annex  to  the  multiplicand  2 ciphers  if  the  multiplier  is  99  or  101 ; 
3 ciphers  if  999  or  1001,  etc.  To  this  result  add  the  multiplicand  when  the  multi- 
plier is  1 more,  or  subtract  when  the  multiplier  is  1 less ; if  2 more,  add  twice  the 
multiplicand  ; if  2 less,  subtract  twice  the  multiplicand,  etc. 


30 


MULTIPLICATION 


WRITTEN  EXERCISE 


81.  1.  Multiply  $1875.24  by  97. 

2.  Multiply  $234.70  by  99. 

3.  Multiply  $558.96  by  998. 

If.  Multiply  $437.10  by  95. 

5.  Multiply  $442.37  by  997. 


6.  Multiply  $120.92  by  9997. 

7.  Multiply  $316.75  by  96. 

8.  Multiply  $242.71  by  999. 

9.  Multiply  $716.32  by  94. 
10.  Multiply  $176.19  by  9999. 


82.  To  multiply  when  one  part  of  the  multiplier  is  a factor  of  the 
other  part. 

Examples. — (1)  Multiply  $325.50  by  357  ; (2)  7854  by  428. 


/ 3 2 3~.  SO  y p ^ 

7 22.  r 

2 2 7 f cT  O Product  by  7 3 / ^3  / Product  by  4. 

/ / 3 3 2 S'  3 “ “ 5X7.  2/33/2“  “ 7X4. 

///  2-  O 3 . S O 3 3 / 2 


83.  Rule. — First  multiply  by  one  factor  ; then  multiply  the  partial  product  by 
the  other  factor. 

Note. — The  factors  of  a number  are  the  numbers  which  when  multiplied  will  produce  it  ; as  7 and 
4 are  the  factors  of  28. 


WRITTEN  EXERCISE 


84.  1.  Multiply  28317  by  284. 

2.  Multiply  7645  by  328. 

3.  Multiply  3624  by  324. 
F Multiply  9463  by  637. 

5.  Multiply  3785  by  426. 


6.  Multiply  3524  by  735. 

7.  Multiply  2835  by  636. 

8.  Multiply  9072  by  954. 

9.  Multiply  648  by  432. 

10.  Multiply  311035  by  864. 


85.  To  multiply  by  11,  111,  etc. 

Example. — Multiply  193  by  11,  the  result 
by  111,  and  that  result  by  1111. 

86.  Rule. — To  multiply  by  11,  set  down  the 
right-hand  figure  of  the  multiplicand,  then  the  sum  of 
the  first  and  second  figures,  the  sum  of  the  second  and 

third,  and  so  on,  carrying  as  usual,  and  writing  the  / / / / 

left-hand  figure  of  the  multiplicand  (plus  any  carrying  2 / / F / 0 aP  F'  3 

figure)  as  the  last  figure  or  figures  of  the  product. 

To  multiply  by  111,  set  doivn  the  right-hand  figure  of  the  multiplicand  as  before, 
then  the  sum  of  the  first  and  second,  then  the  sum  of  the  first,  second  and  third,  of  the 
second,  third  and  fourth,  and  so  on,  carrying  as  usual.  For  1111,  use  each  figure 
four  times. 


/ f 3 
/ / 
2-  / IF  3 

/ / / 

2-  3 3~  & 3~  3 


SHORT  METHODS 


31 


WRITTEN  EXERCISE 


1. 

3732 

X 

11, 

111. 

7. 

22046 

X 

11, 

111. 

2. 

3937 

X 

11, 

111. 

8. 

15432 

X 

11, 

111. 

3. 

10936 

X 

11, 

111. 

9. 

3215 

X 

11, 

111. 

4- 

26417 

X 

11, 

111. 

10. 

2679 

X 

11, 

111. 

5. 

28375 

X 

11, 

111. 

11. 

2150 

X 

11, 

111. 

6. 

3527 

X 

11, 

111. 

12. 

1728 

X 

11, 

111. 

CROSS  MULTIPLICATION 


88.  Cross  Multiplication  consists  in  multiplying  one  number  by  another 
without  writing  partial  products,  setting  down  the  result  only,  performing  all  of 
the  calculations  mentally.  It  is  especially  useful  in  bill  work.  By 'practise, 
operations  can  be  performed  readily  with  multipliers  of  three  or  more  figures. 

89.  To  multiply  two  figures  by  two  figures,  writing  product  only. 


Example. — Multiply  46  by  32. 


l/  6, 
3 2 
/ //  7 2 


“ 2 times  6 are  12  ” — write  2,  and  carry  1.  “ 3 times  6 are  18  and  1 (carried) 

makes  19  ; 2 times  4 are  8,  plus  19  makes  27  ” — write  7,  and  carry  2. 

“ 3 times  4 are  12,  and  2 (carried)  makes  14.” 


90.  Rule. — Multiply  units  by  units  and  set  down  right-hand  figure.  Multiply 
units  of  multiplicand  by  tens  of  multiplier,  add  the  carried  figure,  and  to  the  sum  add 
product  of  tens  of  multiplicand  by  units  of  multiplier,  setting  down  right-hand  figure. 
Multiply  tens  by  tens,  add  the  carried  figure,  and  write  both  figures  of  result. 


WRITTEN  EXERCISE 


91.  1.  27  X 27  = ? 6.  39  X 74  = 

2.  49  X 26  = ? 7.  48  X 34  = 

3.  33  X 63  = ? 8.  93  X 89  = 

If..  57  X 75  = ? 9.  72  X 67  = 

5.  37  X 86  = ? 10.  64  X 53  = 


? 11.  28  X 35  = ? 16.  37  X 43  = ? 

? 12.  52  X 29  = ? 17.  79  X 38  = ? 

? 13.  45  X 73  - ? 18.  54  X 83  = ? 

? U.  68  X 56  = ? 19.  87  X 94  = ? 

? 15.  47  X 69  = ? 20.  95  X 58  = ? 


92.  To  multiply  three  figures  by  two  figures,  writing  product  only. 

Example. — Multiply  436  by  78. 

8X6=48 — write  8. 

436X78=34008  7X6=42,  plus  4,  46  pins  (8X3)  24=70 — write  0. 

7X3=21,  plus  7,  28  plus  (8X4)  32=60 — write  0. 

7X4=28,  plus  6=34 — write  34. 

93.  R ule. — 1.  Multiply  the  units  {of  the  multiplicand)  by  units  {of  the  multi- 
plier), write  right-hand  figure. 

2.  Multiply  units  by  tens,  add  carried  figure,  then  tens  by  units,  add,  write 
right-hand  figure. 

3.  Multiply  tens  by  tens,  add  carried  figure,  then  hundreds  by  units,  add,  write 
right-hand  figure. 

If..  Multiply  hundreds  by  tens,  add  carried  figure,  write  result. 


32 


MULTIPLICATION 


WRITTEN  EXERCISE 


94.  2.  234  X 46  = ? 6.  613  X 28  = 0 27.  845  X 64  = ? 16.  282  X 48  = 9 


2.  368  X 32  = ? 7.  438 

3 421  X 26  = ? 8 347 

£ 732  X 34  = ? 9.  764 

5.  525  X 72  = 9 10.  956 


47  = ? 12.  529  X 52 
63  = ? 13.  672  X 74 
78  = ? U-  438  X 65 
59  = •?  15.  124  X 57 


? 17.  375  X 37  = ? 
? 18.  913  X 82  = ? 
? 19.  742  X 94  = ? 
? 20.  412  X 76  = 9 


95.  To  multiply  three  figures  by  three  figures,  writing  product  only. 

Example.— Multiply  521  by  346. 

1X6  = 6. 

(lX4)  + 0 carried  +(2X6)  =16. 

521 X 346=1 80266  dX3)+i  “ +(2X4)+(5X6)=42. 

(2X3)+4  “ +(5X4)  =30. 

(5X3) +3  “ =18. 

96.  Rule. — 1.  Units  X units. 

2.  (Units  X tens ) + (tens  X units.) 

3.  (Units  X hundreds ) +-  (tens  X tens)  + (hundreds  X units.) 
4..  (Tens  X hundreds)  + (hundreds  X tens.) 

5.  Hundreds  by  hundreds. 


WRITTEN  EXERCISE 


97.  1.  329  X 642  = ? 

2.  473  X 567  = ? 

3.  234  X 319  = ? 
I 748  X 325  = ? 
5.  456  X 538  = ? 


6.  918  x 475  = ? 

7.  237  X 646  = 9 
<5.  786  X 372  = ? 

9.  589  X 716  = ? 

10.  745  X 824  = ? 


98.  To  square  a number  of  two  figures. 


11.  678  X 343  = ? 

12.  834  X 937  = ? 

13.  712  X 846  = ? 
14-  576  X 672  = ? 
15.  483  X 519  = ? 


Examples. — (2)  Find  the  square  of  42  ; 

(2)  of  64. 

42  (2  X 2) 

64 

(4  X 4) 

42  (4  + 4)  X 2 

64 

(6  + 6)  X 4 + 1 

1764  (4  X 4)  + 1 

4096 

(6  X 6)  + 4 

99.  Rule. — (7)  Multiply  together  the  rigth-hand  figures  ; (2). multiply  the  sum 
of  the  left-hand  figures  by  the  right-hand  figure  of  the  multiplier  ; ( 3 ) multiply  together 
the  left-hand  figures ; carry  at  each  step,  as  in  ordinary  multiplication. 


100.  To  find  the  product  of  two  numbers  whose  units  add  to  10  and 


whose  tens  are  alike. 

Examples. — (2)  Multiply  66  X 64; 
66  6 X 4 — write  24 

.64  '6  + 1 X 6 — write  42 

4224 


(2)  88  X 82. 

88  8 X 2 — write  16 

82  S + 1 X S — write  72 
7216 


101.  Rule. — Multiply  together  the  units  and  ivrite  the  result ; add  1 to  the  tens 
of  the  multiplicand,  multiply  by  the  tens  of  the  multiplier,  and  prefix  the  result  to  the 
product  of  the  units. 


SHORT  METHODS 


33 


WRITTEN  EXERCISE 


1. 

45 

X 

45 

6. 

36 

X 

36 

0. 

95 

X 

95 

/V 
/ . 

56 

X 

54 

3. 

27 

X 

27 

8. 

38 

X 

32 

s 

44 

X 

44 

9. 

67 

X 

63 

5. 

64 

X 

64 

10. 

85 

X 

85 

SLIDING  METHOD 


103.  The  figures  of  the  multiplier  are  written  on  a separate  piece  of  paper 
and  moved  from  right  to  left  or  left  to  right  as  indicated  in  examples.  The 
figures  in  either  the  multiplicand  or  the  multiplier  must  be  written  in  reverse 
order.  If  figures  in  the  former  are  reversed,  the  multiplication  will  proceed  from 
left  to  right ; if  the  latter,  from  right  to  left. 

At  the  first  step  only  one  figure  is  multiplied  (units  X units)  then  paper  is 
moved  one  place  to  the  right ; at  the  second,  two  figures  are  multiplied  (tens  X 
units  and  units  X tens)  ; at  the  third,  three  figures,  (hundreds  X units,  tens  X 
tens  and  units  X hundreds) ; and  so  on,  moving  the  paper  a place  each  time, 
multiplying  together  at  each  step  the  vertical  figures  of  the  multiplicand  and 
multiplier,  and  adding  mentally  the  several  products,  carrying  as  usual. 

When  the  left-hand  figure  of  the  multiplier  is  under  the  right-hand  figure 
of  the  multiplicand  (or  vice  versa),  the  last  operation  is  performed  and  both 
figures  of  the  result  written  down. 

Example  1. — Multiply  456  by  144.  (Figures  of  multiplicand  reversed.) 


First  Step  Second  Step  Third  Step  Fourth  Step  Fifth  Step 


b S'  44  b A-  44  6>  .jT  44  b S'  44  b S'  44 

/ //  //  / 44 //  / 44  //  / 44  44  / o4y4 

44  b 44  b b 44  S'  b b 44  b S'  b b S 

6X4=24.  (5X4)+2  + (6X4).  (4>l)+4+(5X4)  + (lX6).  (4X4)+4  + (5Xl).  (4Xl)+2. 

Write  4 Write  6 Write  6 Write  5 Write  6 

Carry  2 Carry  4 Carry  4 Carry  2 Result  65664 

Note. — Any  of  the  foregoing  exercises  may  be  used  for  practise. 

Example  2 — Multiply  456  by  144.  (Figures  of  multiplier  reversed.) 


Fifth  Step 

44  S'  b 

u 44/ 

b S~  b b 44 

Result  65664 


Fourth  Step 

i/bS  b 

44  44  / 

S'  b b 44 

2 Carried 


Third  Step 

44  S'  b 
44  44  / 
b b 44 

4 Carried 


Second  Step 

44^  ^ 

44  44  / 
b 44 

4 Carried 


First  Step 

44  S b 

^44/ 

44 

2 Carried 


34 


MULTIPLICATION 


WRITTEN  PROBLEMS 

104.  1.  What  is  the  cost  of  3456  sheep  at  $6  per  head? 

2.  A man  bought  a farm  containing  495  acres  at  $28  an  acre.  What  was 
the  cost  of  the  farm? 

3.  A farmer  sold  96  bushels  of  wheat  at  57  cents  a bushel.  How  much 
money  did  he  receive? 

J.  At  63  cents  a bushel,  what  will  325  bushels  of  potatoes  cost? 

5.  Find  the  value  of  16  crates  of  eggs,  each  crate  containing  32  dozen  at 
22  cents  a dozen. 

6.  A cotton  factory  contains  245  looms;  if  each  loom  makes  37  yards  of 
cloth  daily,  how  much  cloth  will  the  factory  make  in  70  days? 

7.  There  are  four  fields,  each  containing  585  hills  of  potatoes,  and  every 
hill  averages  12  potatoes.  How  many  potatoes  in  the  four  fields? 

8.  If  a saw-mill  saws  5768  feet  of  boards  in  a day,  how  many  feet  will  it 
turn  out  in  47  weeks  of  6 days  each  ? 

9.  If  ninety-five  men  can  do  a piece  of  wrork  in  45  hours  at  a cost  of  IS 
cents  per  hour,  what  will  the  work  cost? 

10.  A house  requires  4765  shingles  on  each  of  two  sides  of  the  roof.  How 
many  shingles  are  necessary  for  an  operation  of  80  houses? 

11.  A coal  dealer  bought  45  cars  of  coal  containing  25  tons  each.  What 
was  the  cost  of  it  all  at  $4.50  a ton  ? 

12.  A farmer  harvested  42  bushels  of  grain  per  acre  from  a field  of  22  acres. 
What  did  he  receive  for  it  at  95  cents  a bushel? 

13.  Mr.  H.  Humphreys  bought  6 bureaus  at  $S.75  each  ; 3 easy-chairs  at 
$15.32  each  ; 12  dining-room  chairs  at  $4.67  ; 15  mattresses  at  $13.75  : 2 extension 
tables  at  $17.85  ; 4 mirrors  at  $9.79.  What  was  the  total  amount  of  the  bill? 

11/..  James  White  bought  of  John  Black  26  yards  of  cloth  at  $3.25  per  yard  ^ 
147  yards  of  sheeting  at  26  cents  a yard;  28  yards  of  table  linen  at  $1.75  per 
yard;  23  dozen  towels  at  $3.25  per  dozen. 

He  sold  him  27  lambs  at  $7.15  each  ; 15  yearlings  at  $15  each  ; 47  chickens 
at  57  cents  each.  Which  owes  the  other  and  how  much? 

15.  A railway  1326  miles  in  length  was  constructed  at  an  average  cost  of 
$63,275  per  mile.  What  was  the  total  cost  ? 

16.  A carpenter  works  every  day  in  the  year  but  Sundays.  He  receives 
$5  per  day.  How  much  does  he  save  in  one  year  if  his  expenses  amount  to  $12 
per  week  ? 


DIVISION 


105.  Division  is  the  process  of  finding  how  many  times  one  number  is 
contained  in  another. 

106.  The  Dividend  is  the  number  to  be  divided. 

107.  The  Divisor  is  the  number  by  which  the  dividend  is  divided. 

108.  The  Quotient  is  the  number  which  shows  how  many  times  the  divisor 
is  contained  in  the  dividend. 

109.  The  Remainder  is  that  part  of  the  dividend  left  after  dividing,  when 
the  dividend  does  not  contain  the  divisor  an  exact  number  of  times. 

110.  The  Sign  of  Division  is  -5-  , read  divided  by,  and  denotes  that  the 
number  preceding  it  is  to  be  divided  by  the  number  following  it. 

111.  The  Principles  of  Division  are : 

1.  The  divisor  must  be  a number  of  the  same  kind  as  the  dividend  ; for  division 
is  a continued  subtraction  of  the  divisor  from  the  dividend  (the  quotient  showing 
how  many  times  it  is  taken),  and  it  is  evident  that  nothing  could  be  taken  from 
the  dividend  but  a part  of  itself. 

2.  The  quotient  is  an  abstract  number,  because  it  shows  the  number  of  times 
the  dividend  contains  the  divisor. 

3.  If  both  dividend  and  divisor  be  multiplied  or  divided  by  the  same  number , 
the  quotient  of  the  results  obtained  ivill  be  the  same  as  the  quotient  of  the  original 
numbers. 


SHORT  DIVISION 


112.  The  term  “ short  division  ” is  used  when  the  divisor  is  small  enough 
to  perform  the  operation  mentally  and  write  down  only  the  quotient. 


113.  To  divide  by  short  division. 

Examples. — (1)  Divide  32741  by  6 ; (2)  265085  by  12. 


Rule. — Write  the  divisor  on  the 
f,  j 3 2 7 4^  / left  of  the  dividend,  with  a curved 
IT  S line  between  them,  and  draw  a 
horizontal  line  under  the  dividend. 


/ 2 )2  6 

2 2 0^  of/2 


35 


3G 


DIVISION 


Begin  with  the  first  figure  at  the  left  of  the  dividend  (or,  if  that  figure  is  less  than  the 
divisor,  use  the  first  two  figures),  divide  and  write  the  quotient  under  the  figure  divided 

Thus  in  Ex.  1,  since  6 is  not  contained  in  3,  we  say,  “Six  in  32  five  times,  and  2 over,”  and 
write  5 under  the  2. 

When  the  division  is  not  even  or  exact,  prefix  the  remainder  ( mentally ) to  the  next 
figure  of  the  dividend  before  dividing  it. 


Thus  in  Ex.  1,  prefixing  the  remainder  2 to  the  7,  we  say,  “Six  in  27,  four  times,  and  3 over,” 
and  write  4 under  7 ; then  proceeding,  as  before,  we  say,  “Six  in  34  five  times,  and  4 over,  six  in  41 
six  times  and  5 over,”  writing  the  4 and  6 in  their  proper  positions  and  placing  the  last  remainder  5 
over  the  divisor  6 (with  a line  between  them)  at  the  right  of  the  quotient. 

Whenever  a partial  dividend  (except  the  first ) is  less  than  the  divisor,  write  a 
cipher  (0)  in  the  quotient  and  proceed  with  the  division,  including  the  next  figure  in  the 
partial  dividend. 

Thus  in  Ex.  2,  the  second  division  leaves  a remainder  of  1,  which  prefixed  to  the  cipher  in  the 
dividend  makes  10,  which  is  less  than  12,  so  we  write  0 in  the  quotient,  and,  prefixing  the  10  to  the  8, 
we  say,  “ 12  in  108  nine  times  ” — no  remainder.  Then,  12  being  less  than  5,  we  place  another  cipher 
in  the  quotient  and  write  the  remainder  5 after  it,  with  the  divisor  under  it  in  the  form  of  a fraction,  T53. 

Note. — A “ partial  dividend  ” is  that  part  of  the  dividend  used  at  each  of  the  successive  steps  in 
the  operation. 


ORAL  EXERCISE 

114.  1.  At  3 cents  each,  how  many  oranges  can  be  bought  for  36  cents? 

2.  At  8 cents  a yard,  how  many  yards  of  ribbon  cau  be  bought  for  64  cents? 

3.  Gave  $60  for  sheep  at  the  rate  of  $5  a head,  how  many  did  I buy? 

1^.  If  a wheelman  rides  12  miles  an  hour,  how  long  will  it  take  him  to 
travel  96  miles? 

5.  How  many  kegs  of  10  gallons  each  can  be  tilled  from  barrels  containing 
140  gallons? 

6.  How  many  days  will  it  take  a man  to  earn  $96  if  he  receives  86  a day? 

7.  How  many7  bushels  of  wheat  can  a man  buy  for  $84  if  he  pays  86  for  5 
bushels? 

8.  A man  gave  a group  of  boys  56  cents ; if  each  boy  received  14  cents, 
how  many  in  the  group? 

9.  If  coal  is  $5  a ton,  how  many  tons  can  be  bought  for  $95  ? 

10.  If  12  hams  weigh  180  lbs.,  what  is  the  average  weight  of  a ham? 


DIVISION 


11.  Plow  many  are  36  plus  14,  divided  by  5?  25  plus  15,  divided  by  8 ? 32 
plus  18,  less  2,  divided  by  8 ? 

IS.  How  man}r  are  100  minus  12,  divided  by  11  ? 80  plus  2,  minus  5,  divided 
by  7?  144  less  12,  plus  8,  divided  by  14? 

13.  3 times  a number,  plus  5 times  a number,  minus  the  number,  plus  4 
times  the  number,  equals  how  many  times  the  number?  9 times  a number, 
divided  by  3,  multiplied  by  8,  divided  by  6,  multiplied  by  5,  divided  by  4,  equals 
how  many  times  the  number? 

11^.  Think  of  a number,  multiply  it  by  8,  divide  by  4,  divide  by  2,  add  16, 
subtract  the  number  thought  of,  divide  by  8,  and  the  quotient  is  wdiat? 

15.  Think  of  a number,  multiply  it  by  8,  divide  it  by  4,  multiply  by  4, 
divide  by  8,  add  20,  subtract  the  number  thought  of,  divide  by  10,  and  the 
quotient  is  what? 

16.  If  8 tons  of  hay  cost  $200,  what  is  the  value  of  1 ton? 

17.  If  a train  runs  210  miles  in  six  hours,  what  is  its  average  rate  of  speed  ? 

18.  If  a man  buys  $555  worth  of  flour  at  $3  a barrel,  how  many  barrels 
should  he  receive? 

19.  Enough  linoleum  to  cover  a floor  of  320  square  feet  is  sold  for  $9.60 ; 
what  is  the  price  per  square  foot  ? 

SO.  How  many  bushels  of  oats  at  50  cents  a bushel  can  be  bought  for  $18'. 


WRITTEN  EXERCISE 


115.  1.  Divide  79171923  by  2. 

2.  Divide  87682835  by  3. 

3.  Divide  96303847  by  4. 
4-  Divide  15274652  by  5. 

5.  Divide  84205561  by  6. 

6.  Divide  65425622  by  7. 

7.  Divide  82562828  by  8. 

8.  Divide  12412730  by  9. 

9.  Divide  37142897  by  11. 

10.  Divide  31S12719  by  12. 


11.  Divide  3453488739  by  4. 
IS.  Divide  4568392825  by  6. 
13.  Divide  7643912848  by  7. 
11  Divide  5324769214  by  8. 

15.  Divide  6843179157  by  9. 

16.  Divide  7289374673  by  11. 

17.  Divide  8924321866  by  12. 

18.  Divide  382841 1382  by  7. 

19.  Divide  1431287191  by  11. 
SO.  Divide  6897205400  by  12. 


38 


DIVISION 


LONG  DIVISION 

116.  Long  Division  is  applied  when  the  divisor  is  too  large  to  be  handled 
as  a purely  mental  operation — that  is,  when  the  figures  of  the  successive  multi- 
plications and  subtractions  must  be  written  down. 

How  to  divide  a number  by 
long  division. 

Example. — Divide  13279  by  47. 

117.  Rule  . — Write  the  divisor  to  the 
left  of  the  dividend  with  a curved  line 
between  them,  and  another  curved  line  to 
the  right  of  the  dividend  to  separate  the 
quotient  from  it. 

Determine,  by  inspection,  how  many  times  the  divisor  is  contained  in  the  fewest  left- 
hand  figures  of  the  dividend,  and  write  the  number  to  the  right  of  the  dividend. 

Thus,  in  the  accompanying  Example,  since  47  is  not  contained  in  1 or  in  13,  we  take  132  as  the 
first  partial  dividend.  By  comparing  them  we  see  that  the  left-hand  figure  of  the  divisor  (4)  would  he 
contained  in  13  three  times,  hut  by  inspection,  we  see  that  multiplying  by  3 would  give  2 to  carry 
from  the  3 times  7,  which  added  to  3 times  4 would  make  14.  Hence,  we  conclude  that  the  first  quo- 
tient figure  should  be  2,  which  we  place  in  the  quotient  to  the  right  of  the  dividend. 


44  yj  / 3 z 7 *?  ( Z f z 

W 44 

Jf/ 

3 7 6 
/ / f 
f 44 
2 S 

Result,  282-ff. 


Multiply  the  divisor  by  the  quotient  figure  just  found,  and  ivrite  the  product  under 
the  partial  dividend.  Draw  a line,  subtract  and  write  the  remainder  beneath. 

Multiplying  47  by  2 gives  us  94,  which,  subtracted  from  the  partial  dividend,  leaves  38. 


Bring  down  the  next  figure  of  the  dividend  and  place  it  at  the  right  of  the  remain- 
der, making  the  next  partial  dividend. 

Bringing  down  the  7 gives  us  387  for  a partial  dividend,  which,  by  inspection  as  before,  we  can 
see  will  contain  47  eight  times  ; write  8 in  the  quotient. 

Proceed  as  with  first  partial  dividend,  and  continue  until  all  the  dividend  figures 
have  been  brought  down. 

Multiplying  47  by  8 gives  376,  which,  subtracted  from  387,  leaves  11,  to  which  we  annex  the 
last  figure  (9)  of  the  dividend  and  proceed  as  before.  Write  the  remainder,  if  any,  at  the  right  of  the 
quotient  with  the  divisor  below  it. 


118.  Proof. — Multiply  the  quotient  by  the  divisor,  and  add  the  remainder 
to  the  product.  The  sum  should  equal  the  dividend;  or,  in  an  even  division, 
after  casting  out  9’s  in  the  dividend  and  quotient,  multiply  excess  figures  of 
divisor  and  quotient.  The  excess  figure  of  product  must  then  equal  excess 
figure  of  dividend.  If  there  is  a remainder,  its  excess  figure  is  subtracted 
from  the  excess  figure  of  the  dividend  or  that  figure  +9  (if  smaller  than 
remainder  excess),  which  gives  the  excess  figure  of  dividend. 


SHORT  METHODS 


39 


WRITTEN  EXERCISE 


119. 


1. 

Divide  63247  by  53. 

11. 

2. 

Divide  24876  by  89. 

12. 

3. 

Divide  31845  by  67. 

13. 

S 

Divide  27328  by  73. 

If. 

5. 

Divide  486329  by  213. 

15. 

6. 

Divide  675471  by  379. 

16. 

7. 

Divide  39754  by  437 

17. 

8. 

Divide  298546  by  913. 

18. 

9. 

Divide  3875213  by  634. 

19. 

10. 

Divide  4783976  by  719. 

20. 

5254361  by  2483. 
3989768  by  7356. 
42864374  by  8429. 
56893573  by  3067. 
67654826  by  5246. 
32120487  by  4983. 
74S973254  by  35826. 
656438602  by  48375. 
392876348  by  25764. 
487326454  by  384213. 


3 2 7*0= 


<0= 


SPECIAL  RULES  IN  DIVISION 

120.  To  divide  by  10,  100,  1000,  etc. 

Example.— 34837-^100.  J ^ J = .3/ 

121.  Rule. — Cut  off  at  the  right  of  the  dividend  as  many  figures  as  there  are 
ciphers  in  the  divisor.  The  figures  remaining  to  the  left  will  be  the  quotient,  and  the 
figures  cut  off  from  the  right  will  be  the  remainder. 

122.  To  divide  by  any  number 
ending  in  ciphers. 

Example. — Divide  4456328  by 32700. 

Result,  136JLLUL 

123.  Rule. — Cut  off  the  ciphers  at  the 
right  of  the  divisor  and  an  equal  number 
of  figures  at  the  right  of  the  dividend. 

Find  the  quotient  by  dividing  the  re- 
maining figures  of  the  dividend,  by  the  significant  figures  of  the  divisor,  writing  the 
quotient  figure  at  each  step  directly  above  the  right-hand  figure  of  the  partial  dividend. 

To  the  remainder,  after  this  operation,  annex  at  the  right  the  figures  cut  off  from 
the  dividend,  which  gives  the  entire  remainder. 

124.  To  divide  by  25,  etc. 


/ 3 6 

) z/  z/  s 6 3 
3 2 7 

2 r 

/ 


/ r 6 

f F / 

2 2 S'  3 
/ f 4 2 


f / 2 t 


Examples. — (1)  Divide  62438  by  25;  {2)  45659-^-250. 


52:4-4=13.  Remainder. 
Result,  2497^. 


2 v 

'3  r 

2^77 

S 2 

44  s s 7 


/ 7 2 

3 & 

125.  Rule. — Multiply  the  dividend  by  If,  and  divide  the  product  by  100.  ( This 
can  be  extended,  as  in  multiplication,  to  250;  2500,  125,  1250,  etc.) 


40 


DIVISION 


WRITTEN  EXERCISE 


126.  1.  Divide  3248  by  25. 

2.  Divide  4199  by  25. 

3.  Divide  6327  by  25. 
If.  Divide  14832  by  25. 
-5.  Divide  71248  by  25. 


6.  Divide  34862  by  25. 

7.  Divide  41780  by  25. 

8.  Divide  63745  by  25. 

9.  Divide  42150  by  25. 
10.  Divide  37895  by  25. 


127.  To  divide  by  a composite  number. 

Example. — Divide  3479  by  28. 


7)3479 

4)497 

124i 


28=7X4-  Dividing  by  one  factor,  7,  gives  497,  which,  divided  by  the 
other  factor,  4,  gives  the  quotient. 


Example. — Divide  4265  by  64. 


8)4265 

8)533  + 1 Rem. 

66+5  Rem. 

5x8=40,  plus  1=41  True  Rem. 
Result,  66|^. 


64=8X8  Divide  as  above.  The  second 
remainder  is  a part  of  533  times  8 — a number  of 
eighths , and  must  be  multiplied  by  the  first  divisor 
before  adding  to  the  first  remainder,  to  get  the 
whole  remainder. 


128.  Rule — Separate  the  divisor  into  two  factors,  and  divide  the  dividend  by 
one  factor  ; then  divide  the  quotient  by  the  other  factor. 

If  the  first  division  leaves  no  remainder,  write  the  second  remainder  over  the 
second  divisor,  and  place  at  the  right  of  quotient. 

If  both  divisions  leave  remainders,  multiply  the  second  remainder  by  the  first 
divisor,  add  to  first  remainder,  and  write  sum  over  the  entire  divisor,  placing  at  right 
of  quotient. 


WRITTEN  EXERCISE 

16.  Divide  1312432  by  48  (8  X 6) 


129. 

1.  Divide  34826  by  14  (7  X 2). 

2.  Divide  47563  by  15  (5  X 3). 

3.  Divide  57648  by  16  (4  X 4). 

If.  Divide  37849  by  18  (6  X 3) 

5.  Divide  96876  by  21  (7  X 3). 

6.  Divide  134762  by  22  (11  X 2). 

7.  Divide  312843  by  24  (6  X 4). 

8.  Divide  525301  by  27  (9  X 3). 

9.  Divide  626884  by  28  (7  X 4). 

10.  Divide  317783  by  32  (8  X 4). 

11.  Divide  498632  by  33  (11  X 3). 

12.  Divide  653728  by  35  (7  X 5). 

13.  Divide  329849  by  36  (6  X 6). 
If.  Divide  483764  by  42  (7  X 6). 
15.  Divide  959388  by  45  (9  X 5). 


17.  Divide  3298467  by  49  (7  X 7). 

18.  Divide  1128321  by  54  (9  X 6). 

19.  Divide  2557637  by  56  (8  X 7). 

20.  Divide  4887572  bv  63  (9  X 7). 

21.  Divide  5112863  by  64  (8  X S). 

22.  Divide  6768459  by  72  (8  X 9). 

23.  Divide  8384536  by  77  (11  X 7. 

2 If.  Divide  9394281  by  81  (9  X 9). 

25.  Divide  3838633  by  84  (12  X 7). 

26.  Divide  5637344  by  88  (11  X 8). 

27.  Divide  8565787  by  108  (12  X 9). 

28.  Divide  3254063  by  121  (11  X 11). 

29.  Divide  4080362  by  132  (12  xU). 

30.  Divide  6302011  by  144  (12  X 12). 


DIVISION 


41 


WRITTEN  PROBLEMS 

130.  1 . At  $125  each,  how  many  horses  can  be  bought  for  $4250? 

2.  A bin  contains  128560  lbs.  How  many  bushels  of  56  lbs.  each  does  it 
contain  ? 

3.  If  potatoes  yield  46  bushels  per  acre,  how  many  acres  will  be  required  to 
yield  3312  bushels? 

I,..  A clerk  saves  $57  a month.  How  many  months  will  it  require  to  save 
$1311? 

5.  A fruit  grower  received  $1755  for  195  barrels  of  cranberries.  What  was 
the  price  per  barrel? 

6.  How  many  lots  at  $321  each  can  be  bought  for  $772326? 

7.  One  share  of  bank  stock  is  worth  $98.  How  many  shares  can  be  bought 
for  $22050? 

8.  A public  library  has  a yearly  circulation  of  5696600  books.  How  many 
books  are  taken  daily,  if  the  library  is  open  313  days  in  the  year? 

9.  A company  of  547  men  took  equal  shares  in  a mine  valued  at  $705083. 
How  much  money  did  each  man  invest? 

10.  If  a saw-mill  turns  out  1678152  feet  of  boards  in  294  days,  how  many 
feet  are  sawed  daily  ? 

11.  Two  cities  294  miles  apart  are  connected  by  railroad  at  a cost  of  $6845630 ; 
what  is  the  cost  of  one  mile  of  railroad  ? 

12.  A manufacturer  produced  8643759  yards  of  goods  at  a cost  of  129656385 
cents.  What  was  the  cost  of  manufacturing  one  yard? 

13.  The  dividend  is  72987  and  the  divisor  45;  required  the  quotient  and 
remainder. 

Ilf.  The  divisor  is  587  and  the  quotient  8723;  what  is  the  dividend? 

15.  If  28  horses  cost  $3864,  what  is  the  cost  of  one  horse  ? 

16.  A man  left  $2535  to  each  of  his  four  children,  but,  one  of  them  dying, 
the  money  was  divided  among  the  three  living.  How  much  did  each  receive? 


REVIEW  PROBLEMS 

Addition,  Subtraction,  Multiplication  and  Division. 

131.  1.  A merchant’s  bank  balance  on  Saturday  was  $564.32.  On  Monday, 
he  deposited  $365.45  and  checked  out  $260.17  ; on  Wednesday,  deposited  $85.60  ; 
on  Thursday,  deposited  $125  and  checked  out  $468.57;  on  Friday,  deposited 
$93.75.  What  is  his  bank  balance  after  these  transactions? 

2.  A clerk  who  earns  $65  a month,  pays  $25  a month  board,  $150  a year 
for  clothes,  incidental  expenses  $138,  and  invests  the  remainder  in  books  at  $4  a 
volume.  How  many  volumes  can  he  buy  yearly? 

3.  A clerk’s  present  worth  is  $832.  If  his  salary  is  $100  a month,  and  his 
expenses  $14  a week,  how  many  years  must  he  work  to  be  worth  $4000? 


42 


DIVISION 


If..  A is  worth  $525  ; B is  worth  3 times  as  much  less  $80  : C is  worth  8 
times  as  much  as  A and  B.  How  many  horses  valued  at  $180  each  can  they 
buy  with  their  total  money  ? 

5.  Jones  owes  $864.25,  then  pays  $376.15  ; he  then  buys  for  $1514.27  and 
pays  $935.63  ; he  then  buys  for  $564.28  and  pays  $438.20  ; he  now  pays  $1068.75. 
How  much  does  he  still  owe? 

6.  Bought  544  bushels  of  clover  seed  at  $3.85  a bushel  and  paid  for  it  with 
linen  worth  17  cents  a yard.  How  many  yards  did  it  require? 

7.  Bought  a farm  of  135  acres.  Paid  $5130  cash  and  $145  monthly  for  4 
years  and  6 months.  How  much  did  I pay  per  acre  ? 

8.  A manufacturer’s  weekly  sheet  shows  total  product  as  follows:  1st 
week,  5684  yards;  2d  week,  8647  yards;  3d  week,  7863  yards;  4th  w7eek,  6435 
yards  ; 5th  week,  4639  yards  ; sold  24951  yards  at  88  cents  a yard,  and  the 
remainder  at  84  cents  a yard.  What  w7as  the  average  receipt  for  each  yard 
produced  ? 

9.  A lady  owns  a farm  valued  at  $6300,  and  3 stores  valued  at  $15600.  She 
gives  her  daughter  of  the  farm  and  J of  the  value  of  the  stores ; the 
remainder  is  shared  equally  among  5 sons.  How  much  does  the  daughter 
receive  more  than  each  son  ? 

10.  Owning  45  acres  of  land,  I sell  15  acres  for  $2250  ; the  remainder  I sell 
at  $135  an  acre.  I receive  $6000  cash  and  the  balance  in  pigs  at  $5  each.  How 
many  pigs  did  I receive  ? 

11.  Bought  35  barrels  of  pork  at  $26  per  barrel ; 3 barrels  were  damaged  and 
sold  for  $5  per  barrel  less  than  cost ; the  remainder  were  sold  at  a profit  of  83 
per  barrel.  What  was  the  gain  ? 

12.  A farmer  sold  a grocer  50  bushels  of  potatoes  at  68  cents  a bushel;  12 
barrels  of  vinegar  at  $6  a barrel ; 75  bushels  of  apples  at  72  cents  a bushel ; 52 
pounds  of  butter  at  42  cents  a pound,  and  84  dozen  eggs  at  15  cents  a dozen. 
The  farmer  bought  3 barrels  of  sugar,  386  pounds  each,  at  5 cents  a pound  ; 2 
barrels  of  fish  at  $18  a barrel  ; 5 boxes  raisins,  40  pounds  each,  at  12  cents  a 
pound ; 2 cases  prunes,  80  pounds  each,  at  9 cents  a pound  ; 3 barrels  of  coal  oil, 
45  gallons  each,  at  15  cents  a gallon  ; 18  bushels  of  salt  at  27  cents  per  bushel. 
Which  owes  the  other,  and  how  much  ? 

13.  A farmer  buys  a farm  of  110  acres  at  $75  an  acre;  he  pays  $2200  down 
and  the  remainder  in  5 yearly  instalments.  What  was  paid  each  year? 

Ilf.  A builder  finds  the  cost  of  an  operation  of  65  houses  to  be  as  follows : 
land  $325,000,  material  $185,687,  labor  $205,947.  At  what  price  must  he  sell 
each  house  to  gain  $475  a piece? 

15.  In  constructing  a railroad  25  miles  long,  the  grading  cost  $586,742;  10 
bridges  were  built  at  an  average  cost  of  $45S67 ; a tunnel  was  dug  at  a cost  of 
$168,738.  What  was  the  average  cost  per  mile  ? 

16.  In  a business  exchange  125  horses  valued  at  $31200  were  given  for  25 
pianos.  What  was  gained  by  selling  each  piano  for  $1500? 


PROPERTIES  OF  NUMBERS 


132.  A unit  is  one ; as,  one  book,  one  year. 

133.  A number  is  a collection  of  units ; as,  three,  four  boxes,  five  cents. 

134.  Numbers  are  divided  into  integral  and  fractional,  concrete  and  abstract, 
prime  and  composite,  odd  and  even,  similar  and  dissimilar. 

135.  An  integral  number  expresses  whole  things ; as,  six,  three  years. 

136.  A fractional  number  expresses  parts  of  things  : as,  one  half,  one  fourth 
of  an  inch. 

137.  A concrete  number  is  a number  applied  to  particular  objects  ; as,  two 
desks,  seven  apples. 

138.  An  abstract  number  is  a number  considered  apart  from  objects ; as, 
three,  four,  five. 

139.  A prime  number  has  no  exact  divisor,  except  itself  and  one;  as,  two, 
three,  five,  seven. 

140.  A composite  number  is  the  product  of  factors  other  than  itself  and 
one,  and,  consequently,  is  exactly  divisible  by  them  ; as,  6,  8,  9,  15. 

141.  An  odd  number  is  one  that  leaves  a remainder  of  one  when  divided  by 
two ; as,  5,  7,  9. 

142.  An  even  number  is  one  exactly  divisible  by  two ; as,  8,  10,  12. 

143.  Similar  numbers  are  groups  of  the  same  kind  of  units  ; as,  2 feet,  6 
feet,  8 feet. 

144.  Dissimilar  numbers  are  groups  of  different  kinds  of  units;  as,  two 
■oranges,  five  apples. 

145.  The  prime  factors  of  a number  are  the  numbers  which  when 
multiplied  together  will  produce  it.  A prime  factor  is  a factor  that  is  a 
prime  number.  The  factors  of  12  are  2,  2 and  3 ; of  15,  3 and  5. 

146.  The  result  of  taking  the  same  number  two  or  more  times  as  a factor  is 
called  a power;  as  5x5—25,  the  second  power  (called  the  square)  of  five; 
3 X3 X3=27,  the  third  power  (called  the  cube)  of  three.  The  power  is  usually 
indicated  by  a small  figure,  called  the  exponent,  which  is  written  above  and  at 
tlie  right  of  the  factor  ; as,  32  means  3x3  ; 43  means  4X4X4.  The  small  figure 
indicates  the  degree  of  the  power;  as  2s,  the  fifth  power  of  two  ; 54,  the  fourth 
power  of  five. 


43 


FACTORING 

147.  Factoring  is  the  process  by  which  a composite  number  is  reduced  to 
its  factors.  This  is  done  by  division,  the  number  which  we  divide  being  the 
multiple , and  the  divisors,  the  factors. 

148.  To  factor  a number  is  to  find  the  prime  factors. 

Example. — Find  the  prime  factors  of  63. 

3)63 

3)21 

7 3X3X7=63. 

149.  Rule. — Divide  the  number  to  be  factored  by  any  prime  factor  that  will 
exactly  measure  it;  in  like  manner  divide  the  quotient  thus  obtained  and  the  successive 
quotients  until  one  is  obtained  that  is  prime.  The  several  divisors  and  the  prime  quo- 
tient are  the  prime  factors  sought. 

WRITTEN  EXERCISE 

150.  Find  the  prime  factors  of 


1.  27. 

6.  1825. 

11.  3025. 

16.  9020. 

2.  81. 

7.  1890. 

12.  819. 

17.  17017 

3.  132. 

8.  5346. 

13.  1785. 

18.  1009. 

4.  210. 

9.  3465. 

14-  8729. 

19.  971. 

5.  1732. 

10.  7007. 

15.  1287. 

20.  1063. 

DIVISIBILITY  OF  NUMBERS 

151.  The  following  properties  of  numbers  will  be  found  useful  in  reducing 
fractions  to  their  lowest  terms,  in  resolving  numbers  into  their  prime  factors,  etc., 
as  they  serve  to  abridge  the  labor  of  calculation. 

1.  Even  numbers  are  divisible  by  2. 

2.  When  the  sum  of  the  figures  of  a number  is  divisible  by  three,  the  num- 
ber is  divisible  by  3. 

3.  When  4 divides  the  number  expressed  by  the  two  right-hand  figures  of 
a given  number,  the  given  number  is  divisible  by  4- 

4-  When  the  right-hand  figure  of  a number  is  5 or  0,  the  number  is  divisible 

by  5. 

5.  When  an  even  number  is  divisible  by  3,  it  is  divisible  by  6. 

6.  When  8 divides  the  number  expressed  by  the  three  right-hand  figures  of 
a given  number,  the  given  number  is  divisible  by  8. 

7.  When  the  sum  of  the  figures  of  a number  is  divisible  by  9,  the  number  is 
divisible  by  9. 

8.  1001  and  its  multiples,  which  are  usually  easily  recognized,  are  divisible 
by  7,  11,  and  13. 

9.  When  the  sum  of  the  odd  figures  of  a number  equals  the  sum  of  the 
even  figures,  or  when  the  difference  between  these  sums  is  11  or  a multiple  of  11, 
the  number  is  divisible  by  11. 


44 


GREATEST  COMMON  DIVISOR 


152.  A common  divisor  of  several  numbers  is  a number  that  will  exactly 
divide  each  of  them. 

153.  The  greatest  common  divisor  of  two  or  more  numbers  is  the  greatest 
number  that  will  exactly  divide  each  of  them. 

154.  A number  is  only  an  exact  divisor  of  such  other  numbers  as  contain 
all  its  prime  factors. 

155.  To  find  the  greatest  common  divisor  of  two  or  more  numbers 
by  factoring. 

Example. — Required  the  greatest  common  divisor  of  12,  18  and  36. 

( 12=2X2X3 

The  factors  of  < 18=2X3X3 

[ 36=2X2X3X3 

The  factors  common  to  all  are  2 and  3 ; therefore,  their  product,  2X3,  or  6,  is 
the  greatest  common  divisor. 

2)/z  —/  r — a & 

<3)  6 — f — / r 
' 2 - 3 — T 

Explanation. — Since  2 is  an  exact  divisor  of  each  number,  2 is  a factor  of  the  greatest  common 
divisor.  The  quotients  being  divisible  by  3,  3 is  a factor  of  the  greatest  common  divisor.  The  only 
factors  common  to  all  are  2 and  3 ; their  product  is  6,  the  greatest  common  divisor. 


156.  Rule. — Divide  the  numbers  by  any  prime  number  that  exactly  divides  all  of 
them ; proceed  in  the  same  manner  with  the  quotients  ufitil  no  number  can  be  found 
that  will  divide  all  of  the  last  quotients.  The  product  of  all  the  divisors  used  will  be 
the  greatest  common  divisor. 

157.  To  find  the  greatest  common  divisor  when  the  numbers  are 
large  and  not  easily  factored. 

Example  1. — Required  the  greatest  common  divisor  of  437  and  897. 


^23 


2 0 7 
2 0 7 


45 


46 


GREATEST  COMMON  DIVISOR 


Example  2.-Required  the  greatest  common  divisor  of  9388. 16429  and  18776. 

Abbreviated  Method 


9388 

16429 

1 G.  C.  D.  2347  18776  8 

9388 

18776 

1 

7041 

7041 

3 

2347 

7041 

Note. — This  arrangement  simply  avoids  the  repetition  of  figures  and  thus  saves  time. 

158.  Rule. — Divide  the  larger  number  by  the  smaller,  and  each  preceding  divisor 
by  the  remainder  until  there  is  no  remainder.  The  last  divisor  will  be  the  greatest 
common  divisor. 

WRITTEN  EXERCISE 

159.  Find  the  greatest  common  divisor  of 


1. 

24,  32  and  48. 

9. 

81,  135  and  216. 

2. 

53,  63  and  81. 

10. 

390, 

910  and  1365. 

3. 

72,  108  and  120. 

11. 

96,  108,  132  and  156. 

4. 

39,  52  and  65. 

12. 

36,  84,  108  and  144. 

5. 

81,  105  and  135. 

13. 

365, 

511  and  803. 

6. 

210,  315  and  420. 

n. 

102, 

153  and  255. 

7. 

104,  182  and  351. 

15. 

140, 

250,  360  and  270. 

8. 

180,  216  and  276. 

16. 

144, 

256,  192  and  204. 

17. 

A farmer  has  three  strips  of  timber,  80, 

96 

and 

102  feet,  respectiv 

What  is  the  length  of  the  longest  pieces,  all  of  the  same  length,  that  may  be  cut 
from  them  ? 

18.  Three  schools  containing  120,  210  and  360  students,  respectively,  are 
divided  into  classes,  each  containing  the  same  number  of  students.  What  is  the 
greatest  number  of  students  each  class  can  contain,  and  how  many  classes  of 
this  size  are  there  in  each  school  ? 

19.  A man  distributed  $240,  $336  and  $480  among  the  employees  of  3 mills 
in  equal  sums,  the  sums  being  as  large  as  possible.  Required  the  amount  of  the 
equal  sums  and  number  of  employees. 

20.  A triangular  lot,  the  sides  of  which  are  respectively  168,  192  and  180 
feet,  is  to  be  inclosed  with  a tight  board  fence,  7 feet  high,  the  boards  to  run 
lengthwise.  How  many  boards  will  it  take  if  the}''  are  6 inches  wide  and  of  the 
greatest  possible  length  that  will  require  no  cutting? 


LEAST  COMMON  MULTIPLE 

160.  A multiple  of  a number  is  a number  that  is  exactly  divisible  by  it ; 
as,  10  is  a multiple  of  5 ; 12  of  4 ; 9 of  3. 

161.  A common  multiple  of  two  or  more  numbers  is  a number  exactly 
divisible  by  each  of  them.  Thus,  18  is  a common  multiple  of  6,  3 and  2. 

162.  The  least  common  multiple  of  two  or  more  numbers  is  the  least 
number  that  is  exactly  divisible  by  each  of  them.  Thus,  12  is  the  least  common 
multiple  of  3 and  4. 

163.  A common  multiple  of  two  or  more  numbers  contains  all  the  prime 
factors  of  each  of  those  numbers. 

164.  To  find  the  least  common  multiple  of  two  or  more  numbers. 

Example. — Find  the  least  common  multiple  of  16,  18,  24  and  60. 


D,)  / & - / ? -24  - 6 O 
2)f  - 4 - / 2 -3  O 

iw  - y - 6 - / 

13)2  - V - 3 -/^T 
y 2 - 3 - / - S' 


2 X 2 X 2 X 2 X 3 X 3 X or- 
2*x3xx>r  = y20 


165.  Rule. — IT  rite  the  numbers  in  a row  ; draw  a line  underneath,  and,  select- 
ing for  a divisor  the  smallest  prime  number  that  will  exactly  divide  two  or  more  of 
them,  divide  as  many  as  you  can,  bringing  down  below  the  line  those  numbers  which 
cannot  be  divided,  together  with  the  quotients.  Repeat  this  process  with  the  same 
divisor  until  it  will  no  longer  divide  two  of  the  numbers  beloiv  the  line  ; then  take  the 
next  smallest  prime  factor  that  will  divide  two  or  more,  and  proceed  as  before.  Con- 
tinue thus  until  the  numbers  below  the  line  are  prime  to  each  other.  Finally , multiply 
together  the  numbers  beloiv  the  last  line  and  all  the  divisors,  and  the  product  will  be 
the  least  common  multiple. 


WRITTEN  EXERCISE 


166.  Find  the  least  common  multiple  of 


1.  20  and  36. 

2.  21  and  42. 

3.  12,  18  and  24. 
f.  15,  30  and  45. 

5.  16,  24  and  30. 

6.  42,  84  and  105. 

7.  45,  72  and  120. 

8.  54,  90  and  270. 

9.  40,  80  and  120. 


10.  63,  84  and  98. 

11.  45,  60  and  75. 

12.  18,  36,  72  and  120. 

13.  8, 10,  40  and  60. 

If.  27,  36,  45  and  60. 

15.  28,  35,  49  and  70. 

16.  96,  120,  160  and  132. 

17.  36,  60,  120  and  204. 


18.  150,  180.320  and  480. 

19.  What  is  the  shortest  piece  of  goods  that  can  be  cut  into  pieces  of  12  yds., 
18  yds.  and  24  yds.  and  nothing  remain? 

20.  What  is  the  smallest  tract  of  land  that  can  be  exactly  laid  out  in  fields 
of  either  8,  12,  or  16  acres?  How  many  fields  of  each  kind  can  be  made  ? 


47 


CANCELATION 

167.  Cancelation  is  employed  to  shorten  the  work  of  division,  and  consists 
in  casting  out  the  same  factor  from  both  dividend  and  divisor. 

168.  To  cast  out  a factor  of  any  product  is  simply  to  divide  that  product  by 
the  factor  eliminated,  and  to  cast  out  the  same  or  equal  factors  from  both  divi- 
dend and  divisor  is  equivalent  to  dividing  both  dividend  and  divisor  by  the  same 
number ; it  does  not  change  the  quotient.  Thus,  dividing  30  by  10,  or  three  times 
30  by  three  times  10,  or  one-half  of  30  by  one-half  of  10,  all  produce  the  same 
quotient.  (30-^10=3;  90-r-30=3;  15-h5=3.) 

169.  To  find  a quotient  by  cancelation. 

Example. — Divide  8X9X12  by  3X6X10. 

The  factor  3 is  rejected  from  both  dividend 
and  divisor,  by  canceling  the  3 below  the  line 
and  dividing  the  9 above  the  line  by  3,  cancel- 
ing the  9 and  writing  the  quotient  3 above  it. 
The  factor  3 is  again  rejected,  canceling  the  3 
above  the  9 and  dividing  the  6 below  the  line  by 
3,  canceling  the  6 and  writing  2 below  it.  The 
factor  2 is  then  rejected,  canceling  the  2 below  the 
6 and  dividing  the  8 above  the  line  by  2,  cancel- 
ing the  8 and  writing  4 above  it.  The  factor  2 is 
again  rejected,  canceling  the  4 above  the  8 and 
writing  2 above  it,  and  canceling  the  10  below  the 
line  and  writing  5 below  it.  This  leaves  2X12, 
or  24,  above  the  line,  and  5 below  it ; dividing, 
we  have  the  result,  4|. 

170.  Rule. — Write  the  factors  of  the  dividend  above  a line,  and  the  factors  of  the 
divisor  below  it.  Cancel  all  the  factors  which  are  found  in  both  dividend  and  divisor, 
and  divide  the  product  of  the  remaining  factors  in  the  dividend  by  the  product  of  the 
remaining  factors  in  the  divisor. 

WRITTEN  EXERCISE 

171.  1.  Divide  12X15X18  by  4X6X9. 

2.  Divide  15 X 14X22  by  3 X 11X7. 

3.  Divide  18X26X28  by  14x9x13. 

If.  Divide  20x32x48  by  12x10X15. 

5.  Divide  72x36x24  by  18X48X16. 

6.  Divide  56X77X84  by  63x42x33. 

7.  Divide  180X108X270  by  120X18X45. 

5.  Divide  121x132x84  by  144x66x22. 

9.  Divide  75X320X160  by  20X180X150. 

10.  Divide  240X120x350  by  180X75X80. 

11.  Divide  360X146X145  by  365X100X87. 

12.  Divide  1728X195X363  by  144X65X36. 


2 

V J? 

X Jp  X oT  7 


48 


CANCELATION 


49 


WRITTEN  PROBLEMS 

172.  1.  If  10  apples  cost  35  cents,  what  will  14  apples  cost? 

2.  If  7 barrels  of  flour  cost  $84,  what  will  10  barrels  cost? 

3.  A grocer  bought  324  bushels  of  potatoes  at  40  cents  per  bushel,  and  paid 
in  molasses  at  60  cents  per  gallon.  How  many  gallons  did  it  require? 

4-  How  many  sheep  worth  $2.25  per  head  must  be  given  for  12  tons  of  hay 
at  $15  per  ton  ? 

5.  What  must  be  paid  for  shipping  896  pounds  of  iron  at  $6  per  2240  lbs.? 

6.  How  many  bushels  of  wheat  at  90  cents  per  bushel  must  be  exchanged 
for  120  bushels  of  oats  at  45  cents  per  bushel? 

7.  How  many  times  can  a cask,  holding  3 gallons,  be  filled  from  27  bottles 
each  holding  two  pints  ? 

8.  How  many  cords  of  wood  at  $4.50  per  cord  should  be  given  for  18  barrels 
of  flour  at  $5  per  barrel  ? 

9.  A sold  15  barrels  of  pork,  each  containing  180  pounds,  at  18  cents  per 
pound,  and  received  in  payment  a number  of  hogsheads  of  molasses,  each  con- 
taining 90  gallons,  at  60  cents  per  gallon.  How  many  hogsheads  did  he  receive  ? 

10.  A dairyman  sold  36  firkins  of  butter,  each  containing  56  pounds,  at  25 
cents  per  pound,  and  with  the  money  bought  30  pieces  of  calico  at  8 cents  per 
yard.  How  many  yards  in  each  piece  ? 

11.  How  many  bushels  of  apples  worth  45  cents  per  bushel  must  be  given  in 
exchange  for  60  pieces  of  muslin,  each  containing  48  yards  at  9 cents  per  yard  ? 

12.  A milkman  has  20  cows,  which  give  daily  6 quarts  each,  which  he  sells 
at  five  cents  per  quart.  How  many  hams,  each  weighing  15  pounds,  at  16  cents 
per  pound,  can  he  buy  with  4 days’  milk  ? 

13.  How  many  barrels  of  apples,  each  containing  three  bushels,  at  35  cents 
per  bushel,  must  be  given  for  16  barrels  of  sugar,  each  containing  240  pounds, 
at  7 cents  a pound  ? 

14.  If  a laborer  receives  15  cents  per  hour,  how  many  days  of  10  hours 
each  will  it  require  him  to  earn  the  value  of  6 barrels  of  flour  at  $4.50  per 
barrel  ? 

15.  How  many  crates  of  eggs,  each  containing  30  dozen,  at  IS  cents  per 
dozen,  should  be  given  in  exchange  for  120  rolls  of  paper,  each  containing  15 
yards,  at  32  cents  per  yard  ? 

16.  A dealer  bought  625  pounds  of  cheese  at  12.  cents  per  pound  and 
exchanged  it  for  five  loads  of  corn,  each  load  containing  18  bags,  and  each  bag  3 
bushels.  What  was  the  price  per  bushel  ? 

17.  A cotton  dealer  exchanged  24  bales  of  cotton,  averaging  410  pounds  per 
bale,  worth  12  cents  per  pound,  for  48  loads  of  apples,  each  load  containing  15 
bags  of  2 bushels  each.  What  was  the  price  per  bushel  of  the  apples  ? 

18.  A coal  merchant  exchanged  5 car  loads  of  coal,  each  containing  15  tons, 
at  $4.80  per  ton,  for  5 cases  of  children’s  shoes  at  $3.  How  many  pairs  in  each 
case? 


COMMON  FRACTIONS 


173.  A fraction  is  one  or  more  of  the  equal  parts  into  which  a unit  is 
divided  ; as,  one  half  (f),  two  thirds  (-§•). 

174.  A common  fraction  is  one  which  has  a written  denominator,  and  is 
distinguished  from  a decimal  fraction,  which  has  its  denominator  indicated  by 
the  decimal  point. 

175.  Common  fractions  are  divided  into  'proper,  improper,  simple,  compound , 
and  complex. 

176.  A proper  fraction  is  one  whose  numerator  is  less  than  its  denominator  ; 
as,  three  fourths  (f). 

177.  A n improper  fraction  is  one  whose  numerator  is  equal  to  or  greater 
than  its  denominator ; as,  five  fifths  (J-),  four  thirds  (-f).  It  is  simply  a fractional 
form  of  a whole  or  mixed  number. 

178.  A simple  fraction  consists  of  a single  fraction  ; as,  five  sixths  (f). 

179.  A compound  fraction  is  a fraction  of  a fraction  ; as,  -f  of  f of  f ; 
1 of  i 

8 U1  5- 

180.  A complex  fraction  is  one  whose  numerator  or  denominator,  or  both, 

31  3 2 

contain  fractions;  as,  ~ * ■ 

4 % x 

181.  The  terms  of  a fraction  are  the  denominator  and  the  numerator ; 

the  denominator  is  the  part  written  below  the  line  and  indicates  the  number  of 
parts  into  which  the  unit  is  divided ; the  numerator  is  the  part  written  above 
the  line  and  indicates  how  many  parts  are  taken. 

182.  A mixed  number  consists  of  a whole  number  and  a fraction ; as,  21 


WRITTEN  EXERCISE 


183.  Express  in  figures  : 

1.  One  half. 

2.  One  third. 

3.  Three  fourths. 

4-  Five  sixths. 

5.  Seven  eighths. 

184.  What  do  the  following 


1 -2- 

1.  3. 

9 3 


4 • 


5.  £ 

T £ 


6.  Eight  ninths. 

7.  Nine  tenths 

8.  Ten  twelfths. 

9.  Eleven  fifteenths. 
10.  Fifteen  sixteenths. 

expressions  mean  ? 

X _9_  71 4 

o.  10.  /.  17. 

6 11  £15. 

U.  12.  o.  33. 


Q 3i 
3 5* 

io-  if 


50 


FRACTIONS 


51 


REDUCTION  OF  FRACTIONS 

185.  Reduction  of  fractions  means  changing  their  form  without  changing 
their  value. 

186.  There  are  five  different  steps  or  operations  in  the  reduction  of  frac- 
tions : (1)  reducing  improper  fractions  to  whole  or  mixed  numbers  ; (2)  reducing 
mixed  numbers  to  improper  fractions ; (3)  reducing  fractions  to  higher  terms  ; 
(4)  reducing  fractions  to  their  lowest  term  ; (5)  reducing  fractions  to  their  least 
common  denominator. 

187.  To  reduce  improper  fractions  to  whole  or  mixed  numbers. 

Example. — Change  to  a mixed  number. 

£ 2 ) 4/  <6  7 
y 2 2 
/ 4 y 

/ 2 f 

/ y Result  14-|-f 

188.  Rule. — Divide  the  numerator  by  the  denominator. 

WRITTEN  EXERCISE 


189.  Reduce  to  whole  or  mixed  numbers  : 


i.  n 

6.  41. 

11  411 

16.  141 

2.  V- 

7.  41. 

12.  4iL. 

-/O'  12  0 5 

■ 6 1 

3 ¥■ 

X 34. 

O.  3 . 

13.  4&1. 

IQ  3967 
10 • 215- 

4-  ¥• 

9.  404. 

n.  w- 

IQ  4867 

±v.  219  . 

5. 

10.  424 

15.  -W. 

ran  6 7 8 9 1 
ZU.  3 8 9 - 

190.  To  reduce  mixed  numbers  to  improper  fractions. 

Example. — Change  Bd^-  to  an  improper  fraction. 


3 

.2  3 
/ 0 2 
& r 
s_ 

y p y Result 


191.  Rule. — Multiply  the  whole  number  by  the  denominator , add  the  numerator 
to  the  product,  and  write  the  sum  over  the  denominator. 


WRITTEN  EXERCISE 


192.  Change  to  improper  fractions  : 

1.  81  5.  7ii.  p.  73if- 

2.  Ill  6.  9fi  10.  89ff-. 

3.  13f.  7.  2311.  u.  IO8J3. 

l 16tV  8.  45fi.  12.  13711. 


13.  227if. 
H.  524f|. 

15.  8971-fl. 

16.  967114. 


52 


FRACTIONS 


193.  To  reduce  to  higher  or  lower  terms. 

194.  Multiplying  or  dividing  both  numerator  and  denominator  of  a fraction 
by  the  same  number  changes  the  form  but  not  the  value  of  the  fraction. 

195.  To  reduce  fractions  to  higher  terms. 


Example. — Change  f to  thirty-sixths. 


7 _ 

f 3 6 


196.  Rule. — Multiply  both  terms  of  the  fraction  by  such  number  as  will  raise 
its  denominator  to  the  required  denominator. 

197.  Change  the  following  by  inspection  : 


1. 

| to  9 tbs. 

11. 

f and  f to  21sts. 

2. 

f to  16ths. 

12. 

f and  f-  to  18ths. 

3. 

f to  15ths. 

13. 

f and  f to  28ths. 

1 

f to  18ths. 

U- 

J and  f to  12ths. 

5. 

f to  28ths. 

15. 

f and  t9t  to  55ths. 

6. 

f to  24ths. 

16. 

f-,  f and  Y2  to  36ths. 

7. 

f to  27ths. 

17. 

f,  f and  to  70ths. 

8. 

ff  to  30ths. 

18. 

f,  i>  A an(i  if  to  48ths. 

9 

ff  to  44th  s. 

19. 

ts>  if  and  if  t0  84ths- 

10. 

fi  to  36th  s. 

20. 

§>  T9>  if  an(^  ri  to  156ths. 

198.  To  reduce  fractions  to  lowest  terms. 

Example. — Reduce  f-f  to  lowest  terms. 


J_L 

<5  if  6 3 


Dividing  both  numerator  and  denominator  successively  by  9 and  2,  we  obtain  the  result  §. 

199.  Rule. — Divide  both  terms  by  their  greatest  common  divisor.  Or  divide 
both  terms  by  such  numbers  as  will  exactly  measure  them  until  the  numerator  and 
denominator  are  prime  to  each  other. 


ORAL  EXERCISE 


200.  Change  by  inspection  to  lowest  terms: 


1 -S- 
1 • 12- 

6.  if. 

n.  M- 

16. 

1 9 
o 7 * 

O 8 
2T* 

'y  18 
/ . 2 4 • 

12.  ff. 

17. 

1 1 

TT2- 

Q 9 
36- 

8 AS. 
° ' 2 4' 

13-  if- 

18. 

132 
1 44' 

/ I2 
4-  T6’ 

0 14 

3~5  ’ 

U.  ff. 

19. 

9 9 
1 0 S' 

X 6 

o.  9. 

10.  if. 

15-  fi- 

20. 

7 

9 1* 

WRITTEN 

EXERCISE 

201.  Change  to  lowest  terms  : 

1 6 6 
16  5- 

'2  7 6 
315' 

11  A3-2 

11-  648' 

16. 

720 

115  5' 

a 35 
/S-  TITS'' 

7 19  2 

' ' 2 16' 

16)  8 64 

9 3 6' 

17. 

1009 
2 0 0 2' 

0 7 4 

5 7 6' 

O 7 6 8 
9 6 0' 

1®  7 8 0 

1092' 

18. 

1134 
1 2 8 7' 

1 1 1 9 

4-  13  3' 

Q 7 9 2 

//  1512 

19. 

12  8 7 

1 0 0 8' 

35  2 8- 

1 7 2 S ' 

c 8 0 
140' 

10  392 

5 0 4' 

ir  8 19 
10  ■ 3 0 25' 

20. 

3025 

17  0 17' 

FRACTIONS 


53 


ADDITION  OF  FRACTIONS 

202.  The  denominator  determines  the  fractional  unit,  and  since  only  like 
units  can  be  added  or  subtracted,  there  must  be  a common  denominator. 


203.  The  least  common  denominator  of  fractions  is  the  least  number  into 
which  all  the  denominators  can  be  evenly  divided.  This  may  often  be  deter- 
mined by  inspection,  as  12  is  the  least  common  denominator  of  ^ 

204.  When  the  common  denominator  is  not  seen  readily,  select  the 
largest  denominator,  observe  whether  the  next  largest  will  divide  it  evenly;  if 
not,  select  the  smallest  number  into  which  each  of  these  two  will  divide  evenly ; 
with  this  number  take  the  third  largest  denominator  and  proceed  in  the  same 
manner  until  you  have  a number  which  all  denominators  will  divide  evenly 
This  number  is  the  least  common  denominator. 


205.  When  the  least  common  denominator  cannot  be  determined  by 
inspection,  it  can  be  found  by  getting  the  least  common  multiple  of  the  denom- 
inators, as  indicated  in  the  example.  This  may  be  abbreviated  by  excluding 
all  denominators  that  will  divide  evenly  into  a larger  denominator. 


206.  To  find  the  least  common  denominator. 


Example. — f,  ¥V 

Regular  Method  Abbreviated  Method 


2)  / b 3 b 

2j  i/  — y — / r 
f — 9 

/ 2 — J 


UJ  / — 3 
x x <?=/*/ 4 


2 X 2 X 2X  2X  f =/*/£/ 


Write  denominators  as  indicated  in 
example  and  divide  in  manner  shown  by 
any  number  that  will  be  contained  in 
two  or  more  evenly.  Repeat  the  division 
of  the  numbers  below  the  line  until  you 
can  no  longer  divide  two  ; then  multiply 
the  divisors  and  numbers  below  the  last 
line.  The  result  will  be  the  least  com- 
mon denominator. 


207.  Find  the  least  common 
inspection : 


5_  _9_  11 
7 ’ 10?  14* 
g>  3 7 7 1 1 

5>  8>  1 0’  12- 

/ 2.  1 _1 5_ 

v-  3i  6i  in  12- 


denominator  of  the  following  fractions  by 


5. 

6. 

7. 

8. 


3 7 3 1 1 

O’  8)  10’  1 2 * 

.3  3_  5.  11 

4>  7’  8’  14- 

1 3 5 5 1 1 

2>  4’  7’  14'  16- 

2 3 5 11  _9_ 

3’  4)  8)  12’  16- 


54 


FRACTIONS 


208.  To  add  fractions. 

Example. — Add  ■§,  f and  §. 
105  L.  C.  D. 


6 3 
0 

-?&- 
/ J3 


/<?3  — / UL 

yoj-  ~ ' /os- 


5X7X3=105,  least  common  denominator. 

Since  it  is  impossible  to  divide  two  of  the  denominators  evenly  by  any 
number  larger  than  1,  the  least  common  denominator  is  obtained  by  simply 
multiplying  the  denominators  together.  Each  fraction  is  reduced  to  required 
parts  by  dividing  the  denominator  into  105  and  multiplying  by  the  numerator. 


209.  Rule. — Find  the  least  common  multiple  of  the  denominators.  Reduce  each 
fraction  to  this  denominator  and  add  the  numerators,  writing  the  suvi  over  the  common 
denominator.  Reduce  the  result  to  its  lowest  terms. 


ORAL  EXERCISE 


210.  Reduce  to  common  denominators  by  inspection  and  add  the  following: 


1.  \ and  \. 

2.  J and  \. 

3.  | and  f. 
b t and  f. 
5.  | and  £. 


6 — and  —— 

u.  5 tutu  10. 

7.  and  f. 

8.  i £ and 

9.  f,  f and  £. 
10.  J and  t52- 


11.  f,  | and  f . 

12.  f and  f. 

13.  f,  i and 

U-  b I and  To- 
ld. f,  f and 


16.  f,  -§  and  f. 

17.  I,  | and 

18.  i and  4. 

1.9.  f,  | and  T6T. 
^9.  T%  and  ^ 


WRITTEN  EXERCISE 


211.  Add  the  following  : 


1. 

3 
7 ’ 

t and  t72- 

ry 

/ . 

f>  1.  TT  and  H- 

2. 

3 

5> 

9 ailU  1 8* 

8. 

TT>  ¥’  1 and  T2- 

3. 

1 

2’ 

T.  T aild  $> 

9. 

7 5 8 o n r5  3 

8’  T’  9 d'llu  1 1- 

4~ 

2 

3' 

f and  -sV 

10. 

11  2 17  a n r]  23 
1 8 ’ ¥>  24  t,iiu  36- 

5. 

4 

7’ 

td  A aud  A- 

11. 

65  43  11  5 rmr12 

84’  48’  2D  14  1:111  u T- 

6. 

4 7 8 Q-prl  7 

15’  8>  9 “1JU  54' 

12. 

2 3 8 9 10  ot,^  11 
3’  8’  9’  10’  11  dlla  1 2 

212.  To  add  mixed  numbers. 

Example. — Add  4J,  3§ , 5£  and  4|. 

12  L.  C.  D. 

^ 

3#  = r 

S%  = /0 

-3/  _ (7  Least  common  denominator  of  the  fractions,  12  ; fractions  added 

equal  2§  ; adding  2 in  with  whole  numbers  gives  the  result  18|. 

2 33  = 2% 

/ rfi  / 2 


Note. — Add  the  fractions  separately,  then  combine  the  result  with  the  whole  numbers. 


ADDITION  OF  FRACTIONS 


55 


WRITTEN  EXERCISE 


213.  Add  the  following: 

1.  4f,  3J  and  7^-. 

2.  8§,  7f  and  9J. 

3.  34  6f  and  8^. 
f-  9f,  5f  and  7XV- 

A 6i  7|,  4i  and  84. 

6.  3f,  6f,  8i  and  Ilf 


7.  Hi,  131  gi  and  12*. 
A 154,  7i  3i  and  33f. 

9.  23i,  48|  and  67f. 

10.  87|,  1264  and  168& 

11.  554,  67f  and  83f. 


93-3-,  6Sf  and  75-f. 

IS.  Find  the  sum  of  357f  bu.,  189f  bu.,  697-f  bu.  and  378^-  bu. 

11  Add  897f  ft.,  10S9J  ft.,  8784  ft.,  39541-  ft.  an(j  689f  ft, 

15.  A merchant  had  carpets  in  quantities  respectively  as  follows  : 27Sf  yds., 
348J  yds.,  195f  yds.,  482f  yds.  and  378J  yds.  How  many  yards  in  all? 


WRITTEN  PROBLEMS 


214.  1.  A merchant  sold  from  a piece  of  muslin  16f  yds.,  27f  yds.,  47f  yds. 
and  there  still  remained  unsold  25jf  yds.  How  many  yards  were  in  the  piece? 

2.  A farmer  sold  ten  loads  of  hay  weighing  as  follows  : 37^- cwt.,  454  cwt., 
39fi  cwt.,  38^  cwt.,  47f  cwt.,  314  cwt.,  34f£  cwt.,  41fg  cwt.,  33|-f  cwt.,  50f|  cwt. 
Find  total  weight. 

3.  A bought  f of  a yard  of  cloth  for  $f,  f of  a yard  for  $41  of  a yard  for 

$f,  and  -|  of  a yard  for  How  much  money  did  he  spend? 

f.  A family  burned  in  December  3f  tons  of  pea  coal  and  14  tons  of  nut 
coal  ; in  January,  2f  tons  of  pea  coal  and  1 jf-  tons  of  nut  coal ; in  February,  3f 
tons  of  pea  coal  and  2|  tons  of  nut  coal.  How  much  of  each  kind  was  burned 
during  the  three  months? 

5.  A boy  worked  5f  days  for  S4f,  7J  days  for  $5f,  3f  days  for  $3f,  6§  days 
for  $5^-,  and  Ilf  days  for  $12f.  How  many  days  did  he  work,  and  how  much 
money  did  he  receive  ? 

6.  I purchased  5 hams,  net  weight  of  each,  being  respectively,  17f  lbs.,  23f 
lbs.,  31f  lbs.,  2241  lbs.,  19f  lbs.  Find  total  weight. 

7.  Find  the  weight  of  5 cars  of  coal  of  13f  tons,  lly^-  tons,  12ff  tons, 
14^-  tons,  and  9f  tons. 


8.  Four  cars  of  iron  weighed  respectively,  28^-  tons,  30-^-  tons,  26325-  tons 
and  294  tons.  What  was  the  total  weight? 

9.  A merchant  received  money  in  several  sums,  as  follows : £102J,  £79J, 

£83^-,  £26-2^-,  £5  and  £62f.  What  was  the  sum  of  these  receipts  ? 

10.  Four  wheelmen  rode  in  a day  respectively,  125f  miles,  107f  miles,  89f 
miies  and  95§  miles.  What  was  their  total  mileage  for  the  day  ? 


56 


FRACTIONS 


SUBTRACTION  OF  FRACTIONS 


215.  To  be  subtracted,  fractions  must  have  a common  denominator. 


216.  To  find  the  difference  between  two  fractions. 

Example. — Find  the  value  of  -§ — f. 


63L.C.D. 

Y&3 


217.  Rule. — Reduce  to  a common  denominator,  and  write  the  difference  of  the 
numerators  over  the  common  denominator. 


ORAL  EXERCISE 


218. 

By 

inspection  find  the 

value  of 

1. 

1 

2 

1 

3* 

5. 

9 

1 0 

2 

5- 

9. 

2. 

3 

4 

2 

3* 

6. 

7  

8 

5 

1 2- 

10. 

3. 

5 _ 

6 

1 

2 ’ 

7. 

7  

8 

G 
7 ■ 

11. 

1 

7 

8 

3 

4* 

8. 

1 7 
1 8 

2 

3‘ 

12. 

2 

4 

3 

15- 

1 1 

5 

1 2 

1 6' 

1 7 

3 

2 0 

4’ 

8 

5 

9 

6- 

WRITTEN  EXERCISE 
219.  Perform  the  following  subtractions : 


1.  From  pf  take  f-. 

2.  From  if  take  f . 

3.  From  ff  take  f. 
If.  From  ff-  take  f-. 

5.  From  -ff  takeT5 6 7 82-. 

6.  From  ff  take  -f. 

7.  From  f§  take  -ff. 

8.  From  f-f  take 


9.  From  1f5zg-  take  f. 

10.  From  ^ take  ff. 

11.  From  -fff  take  iff. 

12.  From  ff  take  ff. 

13.  From  ff  take  ff. 

U-  From  fff  take 

15.  From  f§f  take  f. 

16.  From  ff  take  ff. 


220.  To  find  the  difference  between  two  mixed  numbers. 


Example.— Find  the  difference  between  14f  and  3f. 


/7ffY7  = 

= 

/ //y7=y2, 

3'A  = 

37A' 

= 3 

In  practise  it  is  sufficient  3 '/j  74 

to  write : 7 / ^ 

/ 0*/*' 

221.  Rule. — Subtract  the  fractions  and  the  whole  numbers  separately , then  com- 
bine the  results. 

Note. — If  the  fraction  of  the  minuend  is  smaller  than  the  fraction  of  the  subtrahend,  borrow 
one  unit  from  the  whole  number,  reduce  it  to  an  improper  fraction  and  add  it  to  the  fraction  of  the 
minuend. 

WRITTEN  EXERCISE 

222.  Find  the  difference  between 


1.  6-f-  and  2^. 

2.  If  and  f. 

3.  3f  and  If. 
If.  17  and  If. 


5.  13f  and  2f. 

6.  15-f  and  7-f. 

7.  237ff  and  159f . 

8.  587ff  and  258-f. 


9.  67Sif  and  399ff 

10.  1037-^  and  674yf. 

11.  367^  and  287ff. 

12.  973f-  and  5S7^%. 


SUBTRACTION  OF  FRACTIONS 


0/ 


WRITTEN  PROBLEMS 

223.  1.  What  is  the  value  of  (f+f — 1)+§? 

S.  Find  the  value  of  3f+5f-+34 — 2-f. 

3 Find  the  value  of  (16§-f-124) — (5^+6^-) 

What  is  the  value  of  564 — 38§  + 19^ — 84? 

5.  From  the  sum  of  144  and  Ilf  take  the  difference  between  35f  and  264- 

6.  From  take  To3lTT  and  add  14. 

7.  From  a barrel  containing  37f  gallons,  29|  gallons  leaked  out.  How 
much  remains  in  the  barrel  ? 

8.  A farmer  sold  560^-  bu.  from  a bin  containing  927-f  bu.  How  many 
bushels  still  in  the  bin? 

9.  A lady  bought  a shawd  for  $13£,  a hat  for  $12f,  and  a dress  for  $25f,  and 
in  payment  gave  six  ten-dollar  bills.  How  much  change  should  she  receive? 

10.  A man’s  family  expenses  for  the  month  of  January  are:  rent,  $25f, 
clothes,  $17f,  coal,  $4f,  groceries,  $15^4.  If  he  receives  8874  a month,  how  much 
can  he  save  ? 

11.  I bought  9J  tons  of  coal  and  had  it  delivered  in  5 loads,  the  first  con- 
taining If  tons,  the  second  If  tons,  the  third  14  tons,  and  the  fourth  1J  tons. 
What  did  the  fifth  load  contain? 

IS.  A had  375  lbs.  of  wire  and  sold  25f  lbs.  to  B,  75§  lbs.  to  C,  and  37^-  lbs. 
to  D.  How  much  had  he  left  ? 

13  A merchant  sold  from  a piece  of  cloth  37§  yards  to  A,  23f  to  B,  and  12§ 
to  C,  and  there  remain  27/g-  yards  unsold.  Find  number  of  yards  in  the 
piece. 

Ilf.  From  a piece  of  muslin  of  74  yds.,  there  were  sold  to  different  persons  64 
yds.,  84  yds.,  124  yds.,  15f  yds.  and  27f  yds.;  how  many  yards  remain  unsold? 

15.  From  a tract  of  land  of  472|-  acres,  there  were  sold  three  farms  respec- 
tively, 112|  acres,  220f  acres  and  88f-  acres.  How  many  acres  were  retained  ? 

16.  In  locating  a park,  tracts  of  land  were  bought  as  follows : 33|-  acres, 
178§  acres,  82-f-  acres,  13f  acres  and  3f  acres.  Roads  and  building  sites  occupied 
18|  acres,  and  woodlands,  87f  acres;  the  remainder  was  in  grass.  How  many 
acres  in  grass? 

17.  From  25000  bushels  of  wheat,  there  were  taken  four  carloads  of  8374 
bushels,  936J  bushels,  1006f  bushels  and  8834  bushels.  How  many  bushels 
were  left? 

18.  Atrip  of  3230  miles  was  made  as  follows:  4784  miles  the  first  day, 
518f  miles  the  second  day,  490|  miles  the  third  day,  537f  miles  the  fourth  day, 
487|  miles  the  fifth  day  and  427|  miles  the  sixth  day.  How  many  miles  were 
traveled  on  the  seventh  day? 

19.  From  a hogshead  of  molasses  containing  72f  gallons,  were  drawn  9§ 
gallons,  15f  gallons,  13|  gallons  and  27f  gallons.  How  many  gallons  remain? 

SO.  A wheelman  traveled  the  first  day  1264  miles,  the  second  day  137|  miles, 
the  third  day  117f  miles,  the  fourth  day  27f  miles,  the  fifth  day  141  f miles. 
How  many  miles  must  he  ride  to  complete  752^  miles? 


58 


FRACTIONS 


MULTIPLICATION  OF  FRACTIONS 

224.  Multiplication  of  fractions  is  the  process  of  finding  the  product  of  two 
factors  when  one  or  both  of  them  are  fractions. 

Note. — When  any  number  is  multiplied  by  a number  greater  than  1,  whether  a whole  or  mixed 
number,  the  product  is  greater  than  the  multiplicand  ; but  when  a number  is  multiplied  by  a fraction 
less  than  1,  the  product  is  less  than  the  multiplicand. 

225.  To  multiply  a fraction  by  a fraction. 

Example. — What  is  the  cost  of  T7y  of  a yard  of  cloth  at  per  yard? 


-4-  y-Z- 

/ o / X 


& 3 = f2_L 

/ XO  4 0 


02n 


3 


/ o 40 

u 


226.  Rule. — Multiply  the  numerators  for  a new  numerator  and  the  denominators 
for  a new  denominator. 


Note. — The  work  may  often  be  abbreviated  by  cancelation. 

WRITTEN  EXERCISE 


227.  Find  the  product  of 


1. 

2 v 3 

3 y\  4. 

l 

2 4 v 7 
4 9 v9- 

7. 

9 V 6 8 
17-^84- 

10. 

2 V 7 V 3 
X A 2 A -5 . 

2. 

4 V 9 

5 16- 

5. 

1 1 V 4 8 

T6X'5ir- 

8 

6 v 1 

19  v2  4' 

11. 

4 v 3 V 2 8 
9A7A33. 

3. 

7 v 1 2 

8 A 15- 

6. 

3 V 1 1 
11  A 7 5- 

9. 

3 v 5 V 6 

4 A G A 7 . 

12. 

5 V 3 V 9 

6 A 8 A x 5. 

Note. — Compound  fractions  are  connected  by  the  word  of  and  are  simplified  by  the  operation  of 
multiplication. 


Find  the  product  of 
13.  f offXlf. 

U.  f of  * off 
15.  fofHoftfXf 
16  *X«X*of*. 


J'y  4 v 7 8 v 13  2 \ / 1 7 
1 ' • 3 3 A 9 9 A 5 5 5 /N  3 2 * 

18.  -f  of  4 of  ut  j^-. 

19-  tfXtfXtt. 

Of)  4 , 6 S V 1 1 / 1 o 
/SU-  jAj  x 9-  A T3  A T9’ 


228.  To  multiply  a whole  number  by  a fraction. 

Example. — Multiply  57  by  f. 


^7/ 


2 FbjT 

<3  / /? 


02 • 


/9 

£ 


3/% 

3 


229.  Rule. — Multiply  the  whole  number  by  the  numerator,  and  divide  the 
product  by  the  denominator. 


230.  Multiply 
1.  4 by  21. 

2 f by  16. 

3.  4 by  17. 
f I by  26. 


WRITTEN  EXERCISE 

5.  by  38. 

6.  by  68. 

7.  by  25. 

8.  |4  by  75. 


9 ff  by  165. 

10.  by  85. 

11.  66  by  f. 

12.  14  by  Jy. 


MULTIPLICATION  OF  FRACTIONS 


59 


231.  To  multiply  a mixed  number  by  a whole  number. 

Example. — Multiply  229f  by  7. 


5 2 


/ & o / 


7 times  |=^=4f  ; 
write  f-  and  carry  4. 


232.  Rule. — Multiply  the  fraction  by  the  whole  number , adding  the  result  to 
the  product  of  the  whole  members. 


WRITTEN  EXERCISE 

233.  Multiply 

1.  337f  by  9.  5.  397f  by  38. 

2.  426f  by  15.  6.  439!!  by  45. 

3.  568f  by  21.  7.  897f  by  87 

1^.  287-f-  by  35.  8.  989f  by  237. 

234.  To  multiply  a mixed  number  by  a mixed  number 

Example. — Multiply  37J  by  354. 


9.  1378|f  by 

10.  3789fi  by 

11.  4288||  by 

12.  506314  by 


235.  Rule. — Multiply , separately,  the  whole  37X:T 
numbers,  each  fraction  by  the  opposite  whole  number,  (3)  ^35 
and  the  fraction  by  the  fraction  ; add  these  products.  (4)  £X£ 


423. 

629. 

325. 

724. 


3 


/ 


r 3 

/ 

/ 

/ 


/ 3 3 / 


14- 

// 

4/ 


236.  Multiply 

1.  56f  by  271. 

2.  37f  by  25|. 

3.  87f  by  19f. 
I 437f  by  18f. 


WRITTEN  EXERCISE 


5.  178f  by  15f. 

6 267*  by  37|. 

7 468fby35|. 

8.  276H  by  127|. 


9.  468-f  by  2391. 

10.  5671  by  256£. 

11.  6081  by  19711 

12.  763*  by  230*. 


WRITTEN  PROBLEMS 

237.  Find  the  cost  of 

1.  385!  pounds  of  paper  at  2-f  cents  per  pound. 

2.  378f  pounds  of  beef  at  7f  cents  per  pound. 

3.  878f  yards  of  muslin  at  9!  cents  per  yard. 

Jf.  676£  bushels  of  timothy  seed  at  $5.62!  per  bushel. 

5.  173f  dozen  eggs  at  18£  cents  per  dozen. 

6.  375|-  gallons  of  wine  at  $2.37!  per  gallon. 

7.  739f  quarts  of  berries  at  5!  cents  per  quart. 

8.  378T3T  acres  of  land  at  $127-f  per  acre. 

9.  39|-  peeks  of  apples  at  37f  cents  per  peck. 

10.  837f  pounds  of  coffee  at  33f  cents  per  pound. 


60 


FRACTIONS 


DIVISION  OF  FRACTIONS 

238.  Division  of  fractions  is  the  process  of  finding  the  quotient  when 
the  dividend  or  divisor,  or  both,  are  fractions 

Note. — The  quotient  is  larger  than  the  dividend  if  the  divisor  is  less  than  1,  and  smaller  if  the 
divisor  is  greater  than  1. 

239.  The  reciprocal  of  a number  is  1 divided  by  that  number;  as,  the 
reciprocal  of  5 is  i ; of  9 is  f . 

240.  The  reciprocal  of  a fraction  is  1 divided  by  that  fraction,  and 
simply  inverts  the  fraction  ; as,  the  reciprocal  of  f is  f ; of  % is  f 


241.  To  divide  a whole  number  by  a fraction. 

Example. — Divide  2 by  f. 

It  will  be  observed  by  the  following  steps  in 
reasoning  that  the  divisor  naturally  inverts  itself. 
To  divide  by  f is  to  divide  by  J of  2.  If  we  divide 
2 by  2.  we  have  as  a quotient  1.  A divisor  I as 
great  would  produce  a quotient  3 times  as  great, 
that  is,  3 times  1 or  3 ; or  ^ is  contained  3 times 
as  often  as  1,  or  3X2  or  6 times,  and  f is  contained 
2 as  often  as  4 or  £ of  6 times  or  3.  One  may  be  expressed  under  the  whole  number,  and  the  division 
performed  similar  to  a fraction  by  a fraction. 

242.  Rule. — Multiply  the  dividend  by  the  divisor  inverted.  (In  other  words, 
multiply  the  dividend  by  the  reciprocal  of  the  divisor.) 

WRITTEN  EXERCISE 

243.  Divide 

1.  1 by  f.  I 15  by  if  7.  39  by  if. 

2.  9 by  f.  5.  20  by  ff.  8.  45  by  qf. 

3 16  by  f 6.  25  by  ff  9.  168  by  ff. 


10.  224  by  ff. 

11.  375  by  ff. 

12.  670  by  ff. 


2 


_x_ 

/ 


3 

_2_ 

3 


^*4 

OZ‘ 


X 


X 


A. 

2 


3 _ 3 

2 

b — 3 
2 


244.  To  divide  a fraction  by  a whole  number. 

Example  1. — Divide  f by  2. 

x_;_2=f  or  Dividing  the  numerator  divides  the  fraction. 

Example  2 —Divide  f by  3. 

f--:-3=Yy  Or  f-t-f =f  Xf . Multiplying  the  denominator  divides  the  fraction. 

c t \C 

245.  Rule. — Divide  the  numerator  by  the  whole  number,  or  'multiply  the  denom- 
inator by  the  whole  number. 


DIVISION  OF  FRACTIONS 


61 


WRITTEN  EXERCISE 

246.  Divide 

1.  | by  2.  4-  \ by  4.  7.  ff  by  5.  10.  if  by  10. 

2.  f by  3.  5.  m by  9.  8.  44  by  7.  11.  44  by  15. 

8.  1 by  2.  6.  ff  by  8.  9.  & by  9.  12.  ff  by  25. 

247.  To  divide  a fraction  by  a fraction. 

Example. — Divide  f by  f. 

To  divide  by  a fraction,  the  principle  is  the  same, 
| Xf=f  =lf.  whether  the  dividend  be  a whole  number  or  a 

fraction. 

248.  Rule. — Multiply  the  dividend  by  the  divisor  inverted. 


WRITTEN  EXERCISE 

249.  Divide 

1.  f by  f.  4-  f by  f.  7.  ff  by  f.  10.  If  by  f. 

8 i by  f.  5.  if  by  8.  ff  by  f 11.  2f  by  H. 

3 yV  by  f.  d.  by  T9T.  P.  ff  by  yf.  12.  3£  by  If. 

250.  To  divide  a mixed  number  by  a whole  number. 

Example. — Divide  37 J by  2. 


//z  = 


37  divided  by  2 gives  18,  with  a remainder  of  1. 
Adding  this  remainder  to  the  fraction  of  the  divi- 
dend, and  reducing  to  an  improper  fraction,  we 
have  f.  Dividing  § by  the  divisor  we  have  f, 
which  completes  the  quotient. 


251.  Rule. — First  divide  the  integral  part  of  the  mixed  number ; if  there  be  a 
remainder,  add  it  to  the  fractional  part ; then  divide  the  fractional  part. 


252.  Divide 

1.  75J  by  4. 

2.  137f  by  9. 

3.  148-f  by  19. 

4.  267f  by  23. 


WRITTEN  EXERCISE 

5.  3971  f by  27.  9. 

6.  4781f  by  33.  10 

7.  5678f  by  35.  11. 

8.  6438f  by  39.  12. 


87378f  by  129. 
98375fi  by  137. 
876341  by  147. 
93675t4x  by  165. 


253.  To  divide  a whole  number  by  a mixed  number. 

Example. s. — (1)  Divide  243  by  4f ; (2)  7854  by  30J. 


tffej  2 y S tf  _4_  3 Of/¥  = 

/~3 ) 7 2 f 7 ^ 

IH m 7 / 4 

77  -y  f s-i/  y l / _ z y s f _2  s f % 

7 7 / / / 

/~  " 


254.  Rule. — Reduce  both  dividend  and  divisor  to  improper  fractions  having  a 
common  denominator  (by  multiplying  both  by  the  denominator  of  the  fraction  in  the 
divisor),  then  divide  the  numerator  of  the  dividend  by  the  numerator  of  the  divisor. 

Or,  give  the  whole  number  a denominator  of  1,  invert  the  divisor  and  proceed  as 
in  multiplication. 


62 


FRACTIONS 


255.  Divide 

1.  46  by  64. 

2.  32  by  3§. 
79  by  7$. 

I 122  by  9$. 


WRITTEN  EXERCISE 

5.  364  by  Ilf 

6.  892  by  13f. 

7.  478  by  24$f 

8.  572  by  3114. 


9.  1219  by  56fl. 

10.  3924  by  42if 

11.  4685  by  74¥3T. 

12.  62830  by  126$£ 


256.  To  divide  a mixed  number  by  a mixed  number. 

Example. — Divide  358$  by  18f. 


/ rfij  3 j r'A 

/ 7 / 2. 

2 2 S)  U 3 0 0 ^ 

2 2 S 
2 0 S 0 
2 0 2 S 
2 S 
2 2 <S 


Reducing  both  dividend  and  divisor  to  twelfths,  we  divide 
the  numerator  of  the  dividend  by  the  numerator  of  the 
divisor  ; or  reducing  to  improper  fractions,  we  invert  the 
divisor  and  multiply,  canceling  if  possible. 

3 22  f'fi  _ / ffa  = 

x -4-  ...  J 7 & = / ?'/f 

3 f 

3 


257.  Rule. — Reduce  both  dividend  and  divisor  to  improper  fractions  having  a 
common  denominator  (by  multiplying  both  by  the  least  common  multiple  of  the  denom- 
inators of  the  fractions),  then  divide  the  numerator  of  the  dividend  by  the  numerator 


of  the  divisor.  Or,  reduce 
in  midtip li cation. 

258.  Divide 

1.  674  by  121. 

2.  87 f by  16$. 

3.  327f  by  244. 
f 837$  by  19$. 


improper  fractions,  invert 

WRITTEN  EXERCISE 

5.  637$$  by  26$. 

6.  938  by  32f. 

7.  1267 f by  87$. 

67  2376f  by  39f. 


the  divisor  and  proceed  as 


9 6789jf  by  97$4- 

10.  9878  25t  by  104T%. 

11.  9783f  by  129$$. 

12.  24389|  by  214f 


WRITTEN  PROBLEMS 

259.  1.  If  $ of  a pound  of  coffee  cost  24  cents,  bow  much  is  it  per  pound? 

2.  At  $f  per  bushel,  how  many  bushels  of  apples  can  be  bought  for  $S? 

3.  If  ^ of  a pound  of  butter  cost  12|  cents,  what  will  $ of  a pound  cost? 
f At  $j  per  yard,  how  many  yards  of  silk  can  be  bought  for  $74? 

5.  If  a man  saves  each  day,  in  how  many  days  can  he  save  $6f? 

6.  If  a horse  can  eat  f of  a bushel  of  oats  in  one  day,  in  what  time  will  he 
eat  8f  bushels  ? 

7.  If  a train  runs  155$$  miles  in  3|-  hours,  what  is  its  average  rate  per  hour? 


DIVISION  OF  FRACTIONS 


63 


8.  How  many  barrels  of  flour  can  be  purchased  for  $21,  if  f of  a barrel 
cost  $4? 

9.  What  will  3§  tons  of  coal  cost,  if  74  tons  cost  $35? 

10.  If  7 men  can  dig  a cellar  in  3f  days,  in  what  time  can  3 men  dig  it? 

11.  18  men  work  6 days  for  $30,  how  much  should  5 men  receive  for  the 
same  time  ? 

12.  If  a pound  of  butter  cost  18f  cents,  how  much  can  be  bought  for  12J 
cents?  For  904  cents? 

13.  If  74  barrels  of  oil  contain  319f  gallons,  how  many  gallons  will  3f  barrels 
contain  ? 

Ilf..  If  § of  the  value  of  a house  is  $13706,  what  is  the  value  of  |-  of  it? 

15.  If  4J  tons  of  coal  cost  $19f,  what  will  37|  tons  cost?  How  many  tons 
could  be  bought  for  $337 J? 

16.  B bought  a house  and  lot;  the  lot  cost  $3700,  which  was  f of  the  cost  of 
the  house.  What  did  he  pay  for  both? 

17.  If  23f  bags  of  coffee  cost  $410,  what  is  the  cost  of  I2f  bags? 

18.  If  17f  tons  of  hay  cost  $258f,  what  will  he  the  value  of  64  tons? 

19.  If  69f  acres  of  land  cost  $766014r,  what  is  the  value  of  311f  acres  ? 

20.  If  86§  cords  of  wood  cost  $806-^,  what  will  be  the  price  of  914  cords? 

21.  If  28f  tons  of  bran  cost  $718f,  what  will  43f  tons  cost  ? 

22.  How  many  bushels  of  potatoes  at  $f  a bushel  will  pay  for  16f  bushels 
of  wheat  at  $lf  a bushel  ? 

23.  If  If  tons  of  coal  cost  $7§,  how  many  tons  can  he  bought  for  $145f ? 

21f.  If  39f  pounds  of  sugar  cost  $2f,  what  will  72f  pounds  cost  ? 

25.  A man  exchanged  46ff  acres  of  land  worth  $120f  an  acre  for-  land  worth 
$166§  an  acre.  How  many  acres  did  he  receive? 

26.  If  6f  tons  of  hay  cost  $93f-,  how  much  hay  will  be  worth  $157T9g  ? 

27.  A piece  of  muslin  containing  42f  yards  costs  $5f;  what  will  a piece 
containing  63f  yards  cost? 

28.  If  47  f gallons  of  vinegar  are  woi’th  $9T^-,  what  will  be  the  cost  of  5 
barrels  of  43f  gallons  each  ? 

29.  How  many  yards  of  sheeting  worth  19f  cents  a yard  should  be  given 
for  7 hams  weighing  10^-  pounds  each,  at  9ff  cents  a pound? 

30.  How  many  pounds  of  coffee  at  254  cents  a pound  should  he  exchanged 
for  3 rolls  of  butter  weighing  4ff-  pounds  each,  worth  30f  cents  a pound? 


64 


FRACTIONS 


COMPLEX  FRACTIONS 

260.  A complex  fraction  is  simply  an  indicated  division,  and  to  simplify 
it*is  to  perform  the  operation  indicated. 

261.  To  simplify  a complex  fraction. 


Example. — Simplify 


3 

4 

8 


4 S' 


262.  Rule. — Divide  the  numerator  by  the  denominator . 


4 7 ~ 7 


WRITTEN  EXERCISE 


263.  Change  to  simple  fractions: 


2 


o 


S 

5. 

6. 


54 

ry 

/ . 

2 11 

3T  2 

m A + 
1 6 

74  ' 

3  1 * 

4 2 

21 

3f 

8. 

3 1_2 

8 1 3 

■f  i 

2 1 1 

11.  3 1 

7 1 

8 2 

9. 

2|+34-t 

1 

1%  3 

39  ' 

6f — 24+  5f 

4 

5 

8 


I off 


4 of  f 

2 _i_  3 ‘ 

3 


3 ofli 
5 U+2J 

4 . 2 I 

o * 3 


3* 


SHORT  METHODS  IN  THE  OPERATIONS  OF  FRACTIONS 

Addition  and  Subtraction. 

264.  To  add  or  subtract  two  simple  fractions. 

Example. — -f  and  f. 

21  20  41  (7X3)  + (5X4)  41_11S 

£ + f ~ 28  — 128  or  7X4  — 28  28' 

265.  Rule. — Multiply  each  numerator  by  the  opposite  denominator  and  write 
the  sum  or  difference  over  the  product  of  the  denominators. 


Multiplication  and  Division. 

266.  To  multiply  a whole  number  by  a mixed  number  that  is  r+  R, 
H’  etc.,  of  10, 100, 1000,  etc. 


Examples. — (1)  Multiply  256  by  12J;  (2)  1896  by  33+ 

explanation 

(1)  12i=i  of  100. 

(2)  33i=i  of  100. 

267.  Rule. — Annex  to  the  multiplicand  one  cipher  if  the  multiplier  is  a part 
of  10 , two,  if  a -part  of  100,  etc.,  and  divide  by  the  denominator  of  the  fractional  part. 


rj  2 S O O 
3 2-00 


3)  / r f ^ oo 
' £'3  2 0 0 


SHORT  OPERATIONS  OF  FRACTIONS 


65 


268.  Multiply 

1.  432  by  124. 

2.  276  by  8|. 

3 348  by  14$. 
If.  96  by  24. 


WRITTEN  EXERCISE 

5.  960  by  16§. 

6.  593  by  14$. 

7.  878  by  334. 

5.  498  by  Ilf 


9.  325  by  6§. 

10.  750  by  9jf. 

11.  832  by  34. 

12.  1248  by  64. 


269.  If  tlie  part  is  4.  i,  b etc.,  greater  or  less  than  10,  100,  etc.,  add  the 
result  to  or  subtract  from  the  increased  multiplicand  as  indicated  in  examples. 

Examples. — (1)  Multiply  187  by  116§ ; (2)  436  by  91§. 


&>)  / S'  7 0 O 
. 3 / / 4% 

Z / S'  / 6 % 


EXPLANATION. 

(3) 164=4  of  100. 

(4)  84=*  of  100. 


7 41 


a 3 i 0 0 

3_  & 3 


3 f f f c y3 


270.  Rule  . — Add  or  subtract  the  fractional  part  greater  or  less  than  10,  100,  etc. 


271.  Multiply 

1.  574  by  1124. 

2.  872  by  874- 

3.  196  by  116$. 
b 572  by  lllf 


WRITTEN  EXERCISE 

5.  472  by  834. 

6.  784  by  75. 

7.  847  by  114$. 

5.  686  by  85f 


9 384  by  1334. 

10.  784  by  66$. 

11.  1210  by  109JL. 

12.  750  by  125. 


272.  To  divide  a whole  number  by  a mixed  number  that  is  %,  y%, 
of  10,  100,  etc. 

Examples. — (1)  Divide  960  by  124;  (2)  385  by  334- 


f 

6, 

0 

3 

? s 

r 

3 

7 ^ 

r 

/ / 

s s 

CL 


/ O o 


// 
z o 


273.  Rule. — Multiply  the  dividend  by  the  denominator  of  the  equivalent  frac- 
tion. Mark  off  one  right  hand  figure  if  the  divisor  is  a fractional  part  of  10  ; two,  if 
a fractional  part  of  100,  etc.  The  remainder  will  be  tenths  if  a fractional  part  of  10, 
and  hundredths  if  a fractional  part  of  100,  etc. 


274.  WRITTEN  EXERCISE 


1.  Divide  432  by  124- 
2 Divide  276  by  84. 
3.  Divide  595  by  16f. 
If.  Divide  348  by  14$. 


5.  Divide  384  by  334- 

6.  Divide  189  by  124. 

7.  Divide  297  by  94t- 

8.  Divide  1462  by  1334. 


66 


RELATION  OF  NUMBERS 


RELATION  OF  NUMBERS 

275.  Finding  the  relation  which  one  number  bears  to  another  is  a process 
of  division,  and  the  result  is  expressed  in  times  either  integrally  or  fractionally. 

Ex. — What  is  the  relation  of  10  to  5 ? 

Ans. — 10  is  2 times,  or  twice  5. 

Ex. — What  is  the  relation  of  5 to  10  ? 

Ans. — 5 is  one-half  10. 

Note. — Only  numbers  of  the  same  denomination  can  be  thus  compared 

MENTAL  PROBLEMS 

276.  7.  What  is  the  relation  of  8 to  4?  Of  12  to  6 ? Of  36  to  12  ? Of  12 
to  84  ? Of  72  to  6 ? 

2.  What  is  the  relation  of  96  to  8?  Of  11  to  121  ? Of  132  to  12  ? Of  132 
toll?  Of  132  to  6?  Of  8 to  144? 

3.  What  part  of  8 apples  is  6 apples  ? 

f What  part  of  16  bushels  is  10  bushels  ? 

5.  What  part  of  35  gallons  is  20  gallons  ? 

6.  What  part  of  42  yards  is  30  yards? 

7.  What  part  of  75  acres  is  35  acres? 

8.  What  part  of  44  quarts  is  32  quarts  ? 

9.  What  part  of  10  is  24  ? 3J?  5?  If?  If  ? If? 

10.  What  part  of  100  is  12f?  25?  16-|?  14f?  SJ?  9^?  33f?  20?  6f?  111? 
3i?  24 ? 374?  624? 

11.  What  part  of  1000  is  250?  333i?  125?  166f?  83f?  200? 

12.  What  part  more  than  10  is  12|?  13f  ? 15?  What  part  less  than  10  is 

7i?  6| ? 8f? 

13.  What  part  more  than  100  is  125  ? 133f?  1124?  116f?  120?  108f? 

150?  1111?  109^?  1141? 

n.  What  part  less  than  100  is  75  ? 87|  ? 66f  ? 83J  ? 90  ? 85f  ? 8Sf  ? 90if  ? 80? 

75.  6f  is  what  part  of  13i  ? Of  20?  Of33f?  Of66§?  Of  8? 

16.  If  4 yards  of  silk  cost  $12,  what  will  8 yards  cost? 

Solution. — 8 yards  will  cost  twice  as  much  as  4 yards,  or  $24. 

17.  If  10  pounds  of  sugar  cost  50  cents,  what  will  100  pounds  cost? 

18.  If  7 yards  of  chintz  are  worth  91  cents,  what  is  the  value  of  21  yards? 

19.  If  8 pairs  of  shoes  cost  $15,  what  are  32  pairs  worth? 

20.  If  12  oranges  cost  35  cents,  what  is  the  value  of  36  oranges? 

21.  If  15  horses  cost  $800,  what  are  75  horses  worth? 

22.  If  9 men  can  build  17  rods  of  wall  in  a given  time,  how  many  rods  can 
54  men  build  in  the  same  time? 


RELATION  OF  NUMBERS 


67 


23.  A man  bought  11  pigs  for  $9,  how  many  could  he  have  bought  for  $45  ? 
2Jf..  4 men  plowed  28  acres  in  6 days;  how  many  acres  could  they  plow  in 
18  days? 

25.  How  much  will  5 books  cost  if  25  books  cost  $15  ? 

Suggestion. — 5 books  are  1 of  25  books. 

26.  What  cost  7 tons  of  hay  if  28  tons  cost  $168? 

27.  If  a wheelman  rode  510  miles  in  6 days,  how  far  could  he  ride  in  2 
days  ? 

28.  If  84  bushels  of  wheat  are  worth  $77,  what  is  the  value  of  12  bushels? 

29.  If  63  gallons  of  molasses  are  worth  $27,  what  will  7 gallons  cost  ? 

30.  A man  had  $30  and  lost  $25,  what  part  of  the  original  amount  has  he 

left  ? 

31.  A man  had  $35  and  received  $70  additional;  how  does  his  present 
amount  compare  with  the  first  amount  ? 

32.  A man  has  $45.  He  spends  $9.  What  part  of  the  original  amount  has 
he  spent  ? What  part  remains  ? 

S3.  In  a bin  of  60  bushels  of  wheat,  20  bushels  are  damaged.  What  part  of 
the  wheat  is  sound? 


WRITTEN  PROBLEMS 

277.  1.  A contributed  $4500  to  a charitable  institution,  B $7500,  and  C 
$3000.  What  part  of  the  total  did  each  contribute? 

2.  A grocer  bought  560  pounds  of  Rio  coffee  and  440  pounds  of  Java. 
What  part  of  the  whole  amount  is  each  ? 

3.  A merchant  bought  3 rolls  of  carpet.  The  first  contained  360  yards,  the 
second  560  yards,  and  the  third  840  yards.  What  part  of  the  total  in  each  roll  ? 

J.  Jones  has  a capital  of  $2250  and  gains  $750;  what  part  of  his  present 
worth  is  his  gain  ? 

5.  A merchant’s  business  expenses  outside  of  rent  for  a year  amount  to 
$3500  and  his  rent  is  $700;  what  part  of  his  total  expenses  is  his  rent? 

6.  A shipper  sends  goods  over  one  railroad  600  miles  and  over  another  400 
miles ; if  his  freight  charges  are  $30,  how  much  should  each  road  receive? 

7.  In  a shipment  of  grain  there  are  6500  bushels  wheat  and  3500  bushels 
oats  ; what  part  of  the  shipment  is  wheat?  What  part  is  oats? 

8.  The  receipts  from  certain  agricultural  products  amount  to  $3000  from 
hay,  $2500  from  potatoes,  and  $2000  from  straw  ; what  part  of  the  whole  amount 
was  received  from  each  ? 

9.  In  a partnership  A invested  $4000,  B $3000,  and  C $2000  ; what  part  of 
the  gain  should  each  receive? 


68 


RELATION  OF  NUMBERS 


10.  A grocer  mixed  40  pounds  of  Mocha  coffee  with  120  pounds  of  Java 
coffee ; what  part  of  the  mixture  is  Java?  What  part  is  Mocha? 

11.  At  a partnership  closing,  D received  $2000  as  his  share  of  gain,  E 
received  $3000,  and  F $1000;  what  part  of  the  gain  did  each  receive  as  his 
share  ? 

12.  Two  kinds  of  coffee  are  blended  at  the  rate  of  2 pounds  of  one  to  3 pounds 
of  the  other;  how  many  pounds  of  each  in  180  pounds  of  the  mixture? 

13.  The  relation  of  $2500  is  to  a second  sum  as  5 to  9 ; what  is  the  second 
sum  ? 

Ilf..  A merchant’s  assets,  $15200,  are  to  his  liabilities  as  5 to  2 ; what  are  his 
liabilities  ? 

15.  In  a shorthand  school  of  220,  the  ladies  are  to  the  gentlemen  as  7 to  4 : 
how  many  of  each  sex  in  the  school? 

16.  A business  man  owes  three  creditors  $2600  in  the  ratio  of  6,  4 and  3; 
what  is  due  each  creditor? 

17.  A business  man  withdrew  from  bank  $1250,  which  was  f-  of  the  amount 
in  bank.  How  much  had  he  in  bank  at  first? 

18.  I pay  $156  for  a horse,  which  is-  f of  the  amount  paid  for  a carriage ; 
what  did  both  cost  me  ? 

19.  A cycler  wheeled  135  miles,  which  was  ■§■  of  the  distance  he  traveled  by 
steamer;  what  was  the  length  of  his  trip  ? 

50.  If  f of  a vessel  cost  $15000,  what  is  the  value  of  -f  of  it? 

51.  If  T7g-  of  an  estate  is  worth  $101500,  what  would  be  the  portion  of  an 
heir  to  f of  it? 

SS.  Bought  f of  a mill  for  $15250 ; what  is  the  value  of  4 of  it  at  the  same 
price  ? 

S3.  If  ^ of  a factory  is  valued  at  $14014,  what  is  -f-  of  it  worth  ? 

Slf.  If  I spend  -f  of  $15,  the  remainder  is  what  part  of  my  friend’s  money, 
who  has  $25  ? 

55.  A has  $45  and  B has  f as  much  plus  $22;  what  part  of  A's  money 
equals  B’s? 

56.  What  is  the  relation  of  124  to  374?  Of  374  to  124?  What  is  the 
relation  of  25  to  874? 

57.  A man  having  of  a ton  of  coal  sold  § of  it ; what  part  of  a ton 
remained  ? 

58.  A merchant  owned  § of  the  stock  of  a store  and  sold  f-  of  his  interest, 
what  interest  in  the  entire  stock  did  he  retain  ? 

59.  A and  B each  had  of  a ton  of  hay  ; B sold  A f of  his  hayr ; what  part 
of  A’s  ecptals  B’s  after  the  sale? 


REVIEW  PROBLEMS  IN  FRACTIONS 


69 


REVIEW  PROBLEMS  IN  FRACTIONS 

278.  1.  Simplify  -}-■  f • 

f of  6 

2.  Add  27f,  671  38f  and 

3.  What  is  the  value  of  of  ff-f  7-|X3-^  of  yf- ? 

A The  product  of  two  numbers  is  405  and  one  of  them  is  27§.  What  is 
the  other  ? 

5.  There  are  two  numbers  whose  difference  is  49|-  and  one  number  is 
of  the  other;  what  are  the  numbers? 

6.  If  a man  travels  4 miles  in  -f-  of  an  hour,  how  far  would  he  travel  in 
2f  hours  at  the  same  rate  ? 

7.  At  $f-  a yard,  how  many  yards  of  silk  can  be  bought  for  $10f? 

8.  If  48  is  f of  some  number,  what  is  § of  the  same  number  ? 

9.  A gentleman  owning  T7g- of  a vessel  sold  ^ of  his  share;  how  much  of 
the  vessel  does  he  now  own  ? 

10.  If  a person  contracts  to  finish  a piece  of  work  in  27  days,  what  part  of 
it  should  he  have  done  in  12  days  ? 

11.  A merchant  lost  f of  his  capital,  after  which  he  gained  $700  and  had 
$2500  ; what  amount  did  he  lose  ? 

12.  A and  B own  a steamboat ; A owns  yk-  and  B the  remainder  of  the 
interest.  If  $7000  has  been  gained  during  a season,  how  much  should  each 
have  ? 

13.  Divide  $4500  between  two  persons  so  that  one  shall  have  j-  as  much  as 
the  other. 

Ilf..  Suppose  the  cargo  of  a vessel  to  be  worth  2-f  times  the  vessel ; what  is  the 
value  of  the  vessel  if  the  cargo  is  worth  $17500? 

15.  If  8|-  tons  of  hay  cost  $100,  how  many  tons  can  be  bought  for  $125  at 
the  same  rate  ? 

16.  If  of  10  bushels  of  wheat  cost  $7|-,  how  much  will  of  8 bushels  cost 
at  the  same  rate  ? 

17.  A cistern  being  full  of  water  sprung  a leak,  and  before  the  leakage  was 
stopped  ■§  of  the  water  had  run  out,  but  f as  much  as  had  run  out  ran  in.  What 
part  of  the  cistern  had  been  emptied  when  the  leak  was  stopped  ? 

18.  How  many  suits  of  clothes  requiring  5-|  yards  each  can  be  cut  from 
121f  yards  of  cloth  ? 

19.  A tank  has  two  outlet  pipes ; one  can  empty  it  in  5 hours,  the  other 
in  8 hours.  In  what  time  will  the  tank  be  emptied  if  both  be  opened  ? 

20.  A can  do  a job  of  work  in  6 hours  and  B can  do  it  in  5 hours  ; in  what 
time  can  both  working  together  do  it? 

21.  A and  B together  have  $1640.  If  B’s  money  is  equal  to  |-  of  A’s,  how 
much  has  each  ? 

22.  A farmer  sold  6 bales  of  hay,  weighing  respectively,  112f  lbs.,  120f  lbs., 
98f  lbs.,  137f  lbs.,  118f  lbs.,  108-|  lbs. ; find  cost  at  If  cents  per  pound. 


70 


FRACTIONS 


23.  If  17J  bushels  of  potatoes  are  equally  divided  among  7 families  of  3 
persons  each,  how  many  should  be  given  to  each  family  and  how  many  to  each 
person  ? 

2 If..  A man  paid  $35  for  a colt,  f of  which  cost  was  of  the  cost  of  a horse. 
What  did  the  horse  cost? 

25.  A grocer  bought  107-g-  bushels  of  potatoes  and  sold  36f-  bushels.  He 
afterwards  purchased  54f  bushels  of  one  man,  37f  bushels  of  another  and  then 
sold  47|  bushels.  How  many  bushels  had  he  then  remaining? 

26.  What  is  the  total  value  of  307 1 yards  of  prints  at  62  cents  per  yard  ; 
4073  yards  cloth  at  $2.061  ; 3252  yards  cassimere  at  $4.37 2 ? 

27.  Bought  a watch  and  chain  for  $140,  the  chain  costing  i as  much  as  the 
watch.  Find  cost  of  each. 

28.  A speculator  has  4 of  his  money  in  city  lots,  \ in  farm  land,  and  the 
remainder,  $17,230,  in  cash.  Find  amount  invested  in  city  lots  and  farm  land. 

29.  How  many  bushels  of  apples  worth  624  cents  per  bushel  must  a farmer 
exchange  for  125f  yards  of  carpet  at  $1.12  per  yard  and  33^  yards  of  cloth  at  $2| 
per  yard  ? 

30.  From  a barrel  containing  37J  gallons  of  syrup,  a grocer  sold  25f  gallons. 
What  part  of  the  barrel  is  unsold? 

31.  A man’s  income  is  $10f  per  week,  and  his  expenses  are  $8|.  Flow  many 
weeks  will  it  require  him  to  save  $137J? 

32.  An  importer  bought  a number  of  bales  of  silk,  each  containing  157 
yards,  at  $lf  per  yard,  and  sold  it  at  $2f,  gaining  $623.  How  many  bales  did 
he  buy  ? 

S3.  A.  B and  C have  a certain  sum  of  money.  A has  f of  the  sum,  B 4 and 
C the  remainder.  Find  the  sum  each  has,  if  A has  $42. 

3 If.  A jeweler  engaging  in  business  lost  -f-  of  the  sum  invested,  after  which  he 
sold  the  remaining  stock,  gaining  $632,  and  received  $3162.  Find  loss. 

35.  A can  do  a piece  of  work  in  5 days  and  B in  6 days.  In  what  time  can 
they  do  it  working  together? 

36.  A and  B can  do  a piece  of  work  in  12  days;  B can  do  it  alone  in  20 
days.  In  what  time  can  A do  it? 

37.  A Liverpool  merchant  bought  56  bales  of  cotton,  each  containing  3674 
lbs.  at  16}f  cents  per  lb.,  and  sold  it  for  19^f  cents  per  lb.  Find  his  gain  if  the 
freight  amounted  to  $97.50. 

38.  How  many  yards  of  cloth  can  a merchant  buy  for  $244,  if  $f  is  paid  for 
f of  a yard  ? 

39.  A bushel  of  wheat  produces  40  pounds  of  flour.  How  many  bushels 
must  be  taken  to  the  mill  to  produce  300  pounds  of  flour,  if  the  miller  takes 
for  toll  ? 

IfO.  What  is  the  value  of  9 pieces  of  goods  containing,  respectively,  47.  493. 
452,  371,  56 3 , 481,  43 2 , 413,  and  461  yards,  at  62J  cents  per  yard  ? 

Ifl.  A man  bought  f of  321 1 acres  of  land  at  $674  per  acre,  and  sold  i-  of  4 
of  the  land  at  $764  per  acre,  and  the  remainder  at  $75f  per  acre.  Find  his  gain. 


GENERAL  REVIEW  PROBLEMS 


71 


42.  How  many  lemons  would  have  to  be  sold  to  gain  20  cents,  if  bought  at 
the  rate  of  2 for  1 cent  and  sold  at  the  rate  of  5 for  3 cents? 

43.  A man  bequeathed  4 of  his  estate  to  his  wife,  § of  the  remainder  to  his 
son,  and  what  was  still  left  to  his  daughter.  If  the  son’s  share  was  $46754  more 
than  the  wife’s,  what  was  the  value  of  the  estate  and  the  share  of  each? 

44-  A man  purchased  a horse  and  buggy  for  $1267.50,  paying  4 more  for 
the  horse  than  for  the  buggy.  Find  the  cost  of  each. 

45.  A and  B have  $1265.  If  A has  f-  less  money  than  B,  how  much  has  each  ? 

46.  A father  divided  $1260  between  his  son  and  daughter,  giving  the 
daughter  -f-  as  much  as  his  son.  How  much  did  each  receive? 

47.  A lady  spent  4 of  her  monthly  earnings  for  board,  of  the  remainder 
for  clothes,  and  still  has  left  $17J.  What  did  she  receive  per  month? 

48.  In  how  many  days  will  a boy  earn  $130  by  sawing  wood  at  $f  per  cord, 
provided  he  saws  2J  cords  per  day? 

49.  How  many  bushels  of  potatoes,  bought  at  $f  per  bushel  and  sold  at  624 
cents,  will  realize  a gain  of  $25? 

50.  I own  \ of  a mill  worth  $8043  and  sell  f of  my  share.  How  much  do  I 
still  own  and  what  is  its  value? 

51.  If  I own  -f-  of  a property,  and  f-  of  my  share  is  worth  $3955,  what  is  the 
entire  property  worth  ? 

52.  A,  B and  C own  a factory.  A owns  B owns  4 more  than  A.  What 
part  does  C own? 

53.  Divide  $3000  into  two  parts  so  that  one  shall  be  f as  much  as  the  other. 

54 ■ A can  do  a piece  of  work  in  8 days,  B in  12  days,  and  C in  15  days. 

After  A and  B work  together  3 days,  how  long  will  it  require  B and  C to  finish 
the  work  ? 

55.  A grocer  buys  sugar  at  the  rate  of  16  pounds  for  $1  and  sells  it  at  the 
rate  of  12  pounds  for  $1.  How  much  will  he  have  to  sell  to  gain  $1  ? 

56.  A,  B and  C engage  in  trade.  A puts  in  $4000,  B $6000  and  C $5000. 
At  the  end  of  the  year  they  find  they  have  gained  $4500.  How  much  should 
each  receive  ? 

57.  A grocer  bought  f of  a barrel  of  molasses  and  sold  § of  it  for  the  entire 
cost  and  the  remainder  at  cost,  and  gained  by  the  transaction  $24.  Find  price 
per  barrel. 

58.  It  requires  8 minutes  for  the  larger  of  two  pipes,  and  12  minutes  for  the 
smaller,  to  fill  a cistern.  Find  time  required  for  both  together. 

59.  4,  of  the  difference  between  two  numbers  is  882.  If  -f  of  the  larger  is 
equal  to  the  smaller,  what  are  the  numbers? 

60.  A young  man  agreed  to  work  for  a year  for  $600  and  a suit  of  clothes. 
His  employer  discharged  him  at  the  end  of  nine  months,  giving  him  $422.50 
and  the  suit.  What  was  the  value  of  the  suit? 

61.  A lady  spent  $224  for  a dress,  $9f  for  a hat  and  $5f  for  shoes.  She  then 
found  that  she  had  left  f of  the  sum  of  money  she  originally  had.  How  much 
had  she  at  first  ? 


72 


FRACTIONS 


62.  If  t7¥  of  a property  is  worth  $762 1 .30,  what  is  the  value  of  jj  - at  the  same 

7 

rate  ? 

63.  B has  in  bank  $237J.  His  income  is  $6f  per  week  and  his  expenses  $9J. 
How  many  weeks  will  his  reserve  fund  hold  out? 

6If.  A broker  invested  $37056  in  land,  and  sold  it  for  times  its  cost,  gain- 
ing $4  per  acre.  How  many  acres  did  he  buy  ? 

65.  If  § of  —tpof  14  times  a man’s  money  is  $2626,  what  is  4 of  - ^ of  3^  times 

4 I2 

his  money  ? 

66.  A can  mow  a field  in  6 days,  and  A and  B together  in  2J  days.  How- 
many  days  will  it  require  B alone  to  mow-  the  field  ? 

67.  A,  B and  C can  do  a piece  of  work  in  10,  12  and  15  hours,  respectively. 
How  long  will  it  take  all  of  them,  working  together,  to  do  the  work?  If  they 
receive  $45,  how  should  it  be  divided  ? 

68.  A father  divided  $2151  between  his  two  sons,  so  that  f of  what  the 
younger  received  equaled  -f  of  what  the  older  received.  Find  amount  received 
by  each. 

69.  A man  owning  f of  a vessel  sold  -f  of  his  share  for  $21242.  Find  value 
of  the  vessel  and  present  value  of  his  share. 

70.  How  long  will  it  require  to  fill  a cistern  with  a capacity  of  337f  gallons, 
if  there  is  a pipe  discharging  into  it  23f  gallons  per  hour  and  there  is  a leak  by 
which  it  loses  24  gallons  per  hour  ? 

71.  If  a boy  buys  peaches  at  the  rate  of  6 for  4 cents  and  sells  them  at  the 
rate  of  4 for  3 cents,  how  many  must  he  buy  and  sell  to  gain  25  cents? 

72.  I spent  -f-  of  my  money  for  an  overcoat  and  f-  of  the  remainder  for  a 
dress  suit.  If  the  overcoat  cost  $111  less  than  the  suit  of  clothes,  what  was  the 
cost  of  each  ? 

73.  A boy  earned  $5621  in  two  years,  and  § of  what  he  earned  the  first  year 
and  $50  equals  what  he  earned  the  second  year.  Find  sum  earned  each  year. 

74-  B lost  | of  his  money,  and  after  earning  f as  much  as  he  had  lost  he  had 
$237J.  How  much  had  he  at  first? 

75.  A boy  and  a man  receive  for  a year’s  work  $1276.  If  the  boy  can  do  f 
as  much  work  as  the  man,  how  should  the  money  be  divided  ? 

76.  A grocer  bought  a tub  of  butter  containing  84  pounds  at  ISf  cents  per 
pound,  and  sold  -f  of  it  for  $14  and  the  remainder  at  a profit  of  41  cents  per 
pound.  Find  gain. 

77.  A miller  bought  72J  bushels  of  wheat  at  S7J  cents  per  bushel  and  sold 
1 of  it  at  921  cents  per  bushel,  and  the  remainder  at  $1,121  Per  bushel.  Find 
his  gain. 

78.  Bought  768  pounds  of  coffee,  and  sold  sufficient  to  amount  to  $87.50  at 
27J  cents  per  pound.  How  much  is  unsold  ? 

79.  A lady  withdrew  ^ of  her  money  from  bank  and  spent  § of  what  she 
withdrew  for  a dress.  If  the  dress  cost  $24,  how  much  money  has  she  still  in 
bank  ? 


GENERAL  REVIEW  PROBLEMS 


73 


80.  A purchased  325  head  of  horses  and  sold  them  so  that  § of  what  he 
received  for  them  equaled  the  cost.  If  his  gain  was  $9875,  what  did  they  cost 
him  per  head  ? 

81.  If  £ of  a farm  is  worth  $450  more  than  -f  of  it,  how  much  is  the  whole 
farm  worth  ? 

83.  £ of  the  sum  paid  for  a house  is  the  sum  paid  for  a lot.  What  is  the 
cost  of  each  if  together  they  cost  $14014.21  ? 

83.  f of  B’s  money  is  equal  to  £ of  A’s  and  the  sum  of  their  money  is  $19008. 
Find  share  of  each. 

84-  A speculator  bought  a tract  of  land  at  $67£  per  acre.  He  sold  £ of  it  at 
$874  per  acre,  -f  of  the  remainder  at  $76£  per  acre,  and  105f  acres,  or  what  still 
remained,  at  $80£.  Find  his  profit. 

85.  A dealer  expended  $5460  for  grain.  £ for  wheat  at  $1.372  per  bushel,  £ 
for  corn  at  $.60§  per  bushel,  and  the  remainder  for  rye  at  $.89  per  bushel.  How 
many  bushels  of  each  kind  did  he  purchase? 

86.  A and  B each  had  f-  of  a ton  of  hay ; B sold  A § of  his  hay.  What 
part  of  A’s  equals  B’s  after  the  sale  ? 

87.  A’s  farm  consisting  of  325f  acres,  cost  him  $624  an  acre.  He  sold  £ of 
it  at  $67  an  acre,  £ of  it  at  $70  an  acre,  and  the  remainder  at  cost.  Find  his  gain. 

88.  I bought  6 hams,  weighing  respectively,  Ilf  lbs.,  12£  lbs.,  7j4>  lbs.,  9£ 
lbs.,  15|  lbs.,  and  13££  lbs.,  at  11£  cents  per  pound.  I sold  the  first  3 at  15f 
cents  per  pound  and  the  last  3 at  16£  cents  per  pound.  Find  my  gain. 

89.  A dealer  bought  5 cars  of  coal  of  13f  tons,  lly^  tons,  12££  tons,  14T9¥ 
tons  and  9^2^-  tons.  Find  the  total  cost  at  $6.75  a ton. 

90.  From  a chain  546£  yards  long  traces  £ of  a yard  long  were  cut.  How 
many  pairs  were  cut,  and  what  part  of  a yard  remained  ? 

91.  A farm  consisting  of  253f  acres  cost  $62f  an  acre  ; £ of  it  was  sold  at 
$67£  an  acre,  £ of  the  remainder  at  $72£  an  acre  and  the  remainder  at  cost.  Find 
the  gain. 

92.  How  many  lots  containing  2^-  acres  can  be  sold  from  a field  containing 
45f  acres,  and  what  will  the  remainder  be  worth  at  $158  an  acre? 

93.  A dealer  bought  5 turkeys  weighing  respectively,  17§  lbs.,  23T9F  lbs.,  21f 
lbs.,22££  lbs.,  and  19£  lbs.  at  13f  cents  per  pound,  and  sold  them  at  16£  cents 
per  pound.  Find  his  gain. 

94 ■ Sold  £ of  the  difference  between  758£  and  948£  tons  of  coal  for  $1328£. 
What  was  the  price  per  cwt.  ? 

95.  A farmer  sold  to  a merchant  3 hams  weighing  lOf  lbs.,  11|  lbs.,  and 
I2££  lbs.  at  12  £ cents  a pound  ; 12£  dozen  eggs  at  254  cents  a dozen.  He  took  £ 
the  value  of  these  commodities  in  cash  (business  result),  4 of  the  exact  remainder 
in  sugar  at  6£  cents  a pound,  and  the  balance  of  exact  result  in  muslin  at  124 
cents  a yard.  How  much  cash  did  he  receive?  How  many  pounds  of  sugar? 
How  many  yards  of  muslin? 


DECIMAL  FRACTIONS 


279.  A decimal  fraction,  or  a decimal,  is  a fraction  whose  denominator  is 
10  or  a power  of  10 ; as,  TV,  jfa,  ToVo- 

280.  In  decimal  fractions,  a unit  is  divided  into  10  equal  parts,  named 
tenths;  the  tenths  are  divided  into  10  equal  parts  named  hundredths.  In 
descending,  each  order  is  y1-^  part  of  the  preceding  order. 

281.  In  a decimal  the  denominator  is  indicated  by  a point  (.)  called  the 
decimal  point.  Sufficient  ciphers  must  be  prefixed  when  necessary  to  make 
the  places  equal  the  ciphers  in  the  denominator.  Thus,  ^ is  written  decimally 

; lino  -03  ; toVo>  -003. 

282.  The  decimal  point  is  also  called  the  separatrix  when  it  separates 
whole  numbers  and  decimals.  In  United  States  money  it  separates  dollars  and 
cents. 

283.  A pure  decimal  consists  of  decimal  figures  only;  as,  .7,  .45,  .375. 

284.  A mixed  decimal  consists  of  a whole  number  and  a decimal ; as,  6 9, 
24.75,  3.1416. 

285.  A complex  decimal  is  a decimal  which  has  at  its  right  a common 
fraction;  as,  .16§,  .18f,  .083J. 

286.  The  decimal  notation  applies  to  integers  and  decimals,  and  the  pro- 
cesses are  the  same  whether  the  scale  is  ascending  or  descending. 

As  we  ascend  from  units,  each  place  in  the  integral  scale  is  increased  tenfold 
and  in  the  decimal  scale  decreased  in  the  same  ratio,  the  first  place  above  units 
being  tens  and  the  first  place  below  units  being  tenths,  etc. 


DECIMAL  NUMERATION 

287.  The  similarity  and  relation  of  the  periods  and  orders  in  a decimal  to 
those  in  a whole  number  are  shown  by  the  following: 


NUMERATION  TABLE 


CO 


9876543210  . 123456789 


Integral  Periods  and  Orders 


Decimal  Periods  and  Orders 


74 


DECIMALS 


75 


WHOLE  NUMBERS  AND  DECIMALS  COMPARED 

288.  General  Principles : 

1.  Both  whole  numbers  and  decimals  increase  from  right  to  left  and 


decrease  from  left  to  right  in  a tenfold 

Sa.  The  farther  an  integral  fig- 
ure is  from  the  units  order  the  greater 
its  value. 

3a.  Placing  a cipher  after  a whole 
number  multiplies  it  by  10  ; two  ciphers 
by  100,  etc. 

Note. — Ciphers  are  not  properly  placed  be- 
fore whole  numbers. 


ratio. 

Sb.  The  farther  a decimal  figure 
is  from  the  decimal  point  the  less  its 
value. 

3b.  Placing  ciphers  after  a deci- 
mal does  not  change  its  value,  while 
placing  a cipher  between  the  figures 
and  the  decimal  point  divides  it  by  10  ; 
two,  by  100,  etc 


NUMERATION  OF  DECIMALS 

289.  How  to  read  a pure  decimal. 

Example. — .1375  is  read,  “One  thousand  three  hundred  seventy-five  ten- 
thousandths or,  “Thirteen  hundred  seventy-five  ten-thousandths.” 

290.  Rule. — First : Begin  at  the  point  and,  numerate  the  decimal  figures  to 
determine  the  name  of  the  right-hand  order.  The  first  place,  being  one-tenth  of  units, 
is  called  tenths  ; the  second,  hundredths  ; the  third,  thousandths  ; the  fourth, 
ten-thousandths , etc.  Observe  carefully  the  written  expression  in  decimals  and 
integers. 

Second : Read  the  decimal  as  if  it  were  a whole  number,  adding  the  name  of  the 
right-hand  order. 

291.  How  to  read  a mixed  decimal. 

Examples. — (1)  24.75  is  read,  “ Twenty-four  and  seventy-five  hundredths;” 
(2)  354.10865  is  read,  “Three  hundred  fifty-four  and  ten  thousand  eight  hundred 
sixty-five  hundred-thousandths.” 

292.  Rule. — After  numerating  the  decimal  figures,  read  the  whole  number,  fol- 
lowing it  with  “ and”  at  the  separatrix ; then  read  the  decimal  after  the  manner  of 
a pure  decimal. 

Note. — Do  not  read  “ and  ” except  at  the  decimal  point.  Thus,  in  the  last  example  do  not  say, 
“Three  hundred  and  fifty-four  and  ten  thousand  eight  hundred  and  sixty-five,”  etc. 

293.  How  to  read  a complex  decimal. 

Examples. — (1)  Read  ,83-J- ; (2)  83.166f . 

Note. — The  common  fraction  at  the  end  of  a complex  decimal  is  not  to  be  regarded  as  occupy- 
ing a decimal  place,  but  as  being  a part  of  1 in  the  last  place  before  it.  Hence,  the  above  examples 
should  be  read,  (1)  “ Eighty-three  one-third  hundredths  ; ” (2)  “Eighty-three  and  one  hundred  sixty- 
six  two-thirds  thousandths.” 


76 


DECIMALS 


WRITTEN  EXERCISE 


294. 

Write  in 

words  or  read  orally  the  following 

numbers : 

1. 

.5 

8. 

2.5 

15. 

70.06 

2. 

.05 

9. 

7.055 

16. 

349.52433 

3. 

.005 

10. 

.055 

17. 

2000.0025 

V 

.1 

11. 

.1003 

18. 

.2025 

5. 

.15 

12. 

,002o5 

19. 

100.1875 

6. 

.025 

13. 

.000109 

20. 

100.18 

7. 

.77 

n- 

32.10005 

21. 

9.003 

NOTATION  OF  DECIMALS 

295.  A decimal  should  contain  as  many  places  as  there  are  ciphers  in  the 
denominator  of  the  equivalent  common  fraction.  In  the  common  fraction,  it  will 
be  observed  that  the  denominator  is  usually  written  and  may  be  any  number; 
in  the  decimal,  the  denominator  is  simply  indicated  and  mud  be  10  or  some 
power  of  10. 

296.  How  to  write  a decimal. 

Examples — (1)  Write  forty-five  hundredths;  (2)  fifty-six  thousandths;  (3) 
fourteen  hundred  twenty-five  ten-thousandths;  (4)  three  hundred  seventy-five 
hundred-thousandths;  (5)  nineteen  thousand  sixty-four  millionths. 

(1)  .45;  (2)  .056  ; (3)  .1425;  (4)  .00375:  (5)  ,olo064. 

Note. — It  is  important  to  distinguish  between  the  number  part  and  the  name  part-  of  a decimal 
expressed  in  words.  When  such  decimals  as  those  in  the  examples  above  are  recognized  as  45  hun- 
dredths, 56  thousandths,  1425  teD-thousandths,  375  hundred-thousandths,  etc.,  the  correct  writing  of 
them  will  be  an  easy  matter  by  simply  observing  the  following  : 

297.  Rule  — First  write  the  figures  of  the  decimal  as  if  they  comprised  a whole 
number,  then  place  the  point  to  the  left  of  as  many  figures  ( counting  from,  the  right- 
hand)  as  there  would  be  ciphers  in  the  denominator  if  written  as  a common  fraction. 

Verify  the  notation  by  numerating  the  decimal  from  left  to  right,  beginning  at 
tenths. 


WRITTEN  EXERCISE 


298.  Write  the  following  : 

1.  9 tenths. 

2.  27  hundredths. 

3.  83  hundredths. 

125  thousandths. 

5.  65  thousandths. 

6.  1375  ten-thousandths. 


7.  Fourteen  hundredths. 

8.  Twenty-five  thousandths. 

9.  Sixty-six  ten-thousandths. 

10.  Twelve  hundred-thousandths. 

11.  Eighty-eight  millionths. 

12.  Ninety-five  tenths. 


DECIMALS 


77 


13.  72  ten-thousandths.  17.  Nine  and  five-tenths 

lit..  15  hundred-thousandths.  18.  Ninety-nine  ten-millionths. 

15.  95  hundredths.  19.  One  hundred-millionths. 

16.  4175  thousandths.  SO.  Eleven  hundred  eleven  hundiedths. 

SI.  One  hundred  one  thousandths. 

SS.  One  hundred  and  one  thousandth. 

S3.  Fifteen  hundred  fifteen  millionths. 

Sj.  Fifteen  hundred  and  fifteen  millionths. 

25.  Twenty-five  thousand  and  twenty-five  thousandths. 

26.  Three  and  fourteen  hundred  sixteen  ten-thousandths 

27.  Four  and  eiglity-six  hundred  sixty-five  ten-thousandths. 

28.  Two  thousand  thirty-seven  and  two  thousand  thirty -seven  ten- 
thousandths. 

29.  Four  hundred  five  and  five  hundred  four  hundred-thousandths. 

30.  Ninety-nine  million  nine  hundred  ninety  thousand  ninety-nine  and 
nine  hundred  ninety  million  ninety-nine  thousand  nine  hundred  ninety-nine 
billionths. 


REDUCTION  OF  DECIMALS 


299.  To  reduce  a common  fraction  to  a decimal. 

Examples — (1)  Reduce  f 


r/3. 

0 0 0 

3 7 cT 

f/J  .dfr^T 

ft-)  f/c  = ./T/5T 

fe)  = . oj-37-r 


to  a decimal ; (2) 

/ ^7,^ 

h. 

0 0 0 0 

/ 

b 

/ 

4 0 

/ 

i r 

/ 2 0 

/ / 2 

r o 

r o 


3 2 


0 

2 

3 7 or 

h. 

0 

'<? 

o'o  0 

2 

r 

r 

/ 

2 

0 

7 

6 

2 2 4 


/ 

/ 


3 
3 3 


300.  Rule. — Place  the  numerator  to  the  right  of  the  denominator  with  a curved 
line  between  them , as  in  short  division.  Draw  a vertical  line  just  to  the  right  of  the 
numerator  and  cross  it  with  a horizontal  line  either  above  or  below  the  numerator,  as 
shown  in  the  foregoing  examples. 

Divide  by  the  denominator , annexing  ciphers  to  the  numerator  as  required,  and 
place  a quotient  figure  either  above  or  below  each  cipher  annexed.  Continue  the  divi- 
sion until  it  results  in  a pure  decimal  or  in  the  number  of  decimal  places  required. 


78 


DECIMALS 


WRITTEN  EXERCISE 


301. 

Reduce  to  decimal  form  : 

1. 

1 

2* 

21. 

111 

12  5- 

32. 

18 

2. 

1 

4* 

13.  18f. 

22. 

249 
2 5 O' 

83J 

3. 

3 

4* 

U.  8*. 

23. 

3 7 7 
T5  O’ 

33. 

15*. 

he- 

5. 

4 

15.  $$. 

16.  37|. 

n. 

25. 

S 1 3 

37$ 

87$ 

5' 

3 

5- 

1 2 5 0' 
80*. 

31 

6. 

9 

2 0- 

17.  16  - 

26. 

18M- 

35. 

r.q  3 
Ddg4. 

7. 

1 3 

2 O' 

m 

27. 

36. 

17  1. 

1 1 1 6 O' 

8. 

1 9 

2 O' 

18.  lo— 

28. 

9tV 

37. 

9.-,  1 
ZOT0- 

9. 

1 3 
25' 

66f 

29. 

131. 

38. 

9,1  3 

Oi4  0 6' 

10. 

3 

8* 

19.  1 ¥5. 

30. 

6|- 

39. 

32*e- 

11. 

7 

8* 

T2*' 

31. 

483V- 

40. 

940*3. 

Note. — Carrying  results  to  four  decimal  places  is  sufficiently  accurate  for  most  commercial 
operations.  The  operation  may  often  be  simplified  by  reducing  the  fraction  to  its  lowest  terms  before 
dividing. 


302.  To  reduce  a decimal  to  a common  fraction. 

Examples. — Reduce  to  common  fractions  (1)  .25  ; (2)  .16| ; (3)  .0625. 

50 


x 25  1 

0) -26  = 100  = 4' 


(3)  .0625  = 


(-)  -16*  = IBo 

625  5 1 


3 X 100 


10000 


o 

80 


16 


303.  Write  the  decimal  in  the  form  of  a common  fraction,  omitting  the  decimal 
point  and  any  ciphers  standing  next  to  it,  and  using  the  figures  for  a numerator  ; for  a 
denominator , write  1 with  as  many  ciphers  annexed  to  it  as  there  are  places  in  the 
decimal.  Reduce  to  lowest  terms. 


Note. — In  the  case  of  a complex  fraction,  simplify  according  to  method  in  common  fractions. 


WRITTEN  EXERCISE 


304.  Change  to  common  fractions  : 


1. 

.25. 

9. 

.16$. 

17. 

3.75. 

25. 

.98$. 

33. 

2.00$. 

2. 

.5. 

10. 

.331. 

18. 

12,5. 

26. 

7.301. 

34. 

.007*. 

3. 

.15. 

11. 

.66$. 

19. 

.0831. 

27. 

2.0875. 

35. 

•6266$. 

4- 

.75. 

12. 

.871 

20. 

3.1416. 

28. 

1.621. 

36. 

3-05*. 

5. 

.125. 

13. 

.061. 

21. 

•57$. 

29. 

3.874. 

37. 

2.0375. 

6. 

.625. 

n. 

.18$. 

22. 

.066$. 

30. 

.5236. 

38. 

6.3178. 

7. 

.875. 

15. 

.Ill 

23. 

•44$. 

31. 

2.007$. 

39. 

21.675. 

8. 

.7854. 

16. 

.14$. 

24. 

.124$. 

32. 

3.0061. 

40. 

44.466$ 

DECIMALS 


79 


ADDITION  OF  DECIMALS 

305.  How  to  add  decimals. 

Example — Add  .5,  .0075,  80.08,  24.75,  3.06,  800  125. 


306.  Rule. — H 7rite  the  numbers  so  that  the  decimal 
points  are  in  the  same  column.  Add  as  in  whole  numbers , 
kerping  the  decimal  point  in  its  proper  place  and  carrying 
as  in  whole  numbers. 

Note. — Whenever  possible,  the  teacher  should  dictate  the 
numbers  to  be  added. 


WRITTEN  EXERCISE 

307.  Add  the  following  : 

1.  .79,  .473,  61.00057,  457.002073. 

2.  81.004,  73.0102,  .08007,  8000.0008,  .8008,  368.5. 

3.  77  ten-thousandths,  751  hundred-thousandths,  319  millionths,  7 
tenths,  7 ten-millionths. 

4-  315  and  75  hundredths,  85  and  85  thousandths,  1523  and  375  ten- 
thousandths,  415  and  1875  hundred-thousandths,  95  and  9 tenths,  2324  and 
17625  millionths. 

5.  Write  the  decimals  from  problem  1 to  problem  10  in  the  exercise  in 
notation  of  decimals  298,  in  proper  form  for  addition,  and  find  the  sum. 

6.  Write  from  11  to  20  in  the  same  exercise  as  a problem  in  addition,  and 
find  the  sum. 

7.  Write  the  numbers  from  21  to  30  in  the  same  exercise  and  add  them. 

8.  Add  the  following  items  of  United  States  money,  placing  the  dollar  sign 
before  the  first  one  only:  $175854.75,  954.32,  1324.56,97.88,  11927.50,  45.12, 
245.29,  64381.95,  19.61,  227433.19. 

9.  Write  the  following  in  a line  and  add  them  horizontally:  4712.25, 

820.75,  5162.50,  112.67,6125.18,231.45,  15.95,  550.08,  9.84,  1243.36.  Prove  result. 

10.  Add  31.135,  15.016,  23.25*,  36  47f,  125.3-f. 

11.  How  many  tons  of  coal  in  4 carloads  weighing  respectively  32.805, 
31.675,  29|,  30.254  tons? 

12.  A merchant  sold  34  yards  of  cloth  for  $5,674,  yards  for  $1.50  and  12.3 
yards  of  another  piece  for  $6,034.  How  many  yards  did  he  sell  in  all  and  for 
how  much  ? 

13.  Find  the  sum  of  175  francs  18  centimes,  38  fr.  96  c.,  115  fr.  45  c.,  24  fr. 
65  c.,  8 fr.  50  c.,  132  fr.  15  c.,  56  fr.  80  c.,  6 fr.  75  c.,  251  fr.  22  c.,  48  fr.  25  c. 

Ilf..  Find  the  sum  of  2150  Marks  55  Pfennigs,  459  M.  75  Pf. , 91  M.  52  Pf., 
1654  M.  25  Pf.,  75  M.  50  Pf.,  353  M.  85  Pf.,  9 M.  95  Pf..  766  M.  20  Pf.,  87  M.  35 
Pf.,  2344  M.  65  Pf. 

Note. — 100  centimes  in  a franc  ; 100  pfennigs  in  a mark. 


.S 

.0  0 7 oT 

r o .o  r 
% 4.7  s 
3.0  & 

<r  0 0 . / x 
f 0 r . s x 2 ur 


80 


DECIMALS 


SUBTRACTION  OF  DECIMALS 


308.  How  to  subtract  decimals. 

Examples. — (1)  From  .875  take  .75  ; (2)  18.75  take  1.375  ; (3)  10  take  .0875 


(i) 

.r  7 s 
JZ.s. 

. / 2 S 


(2) 

/ r .7cT 

/ . 3 7 S 

/7.37s 


(31 

/ 0. 

.o  r 7 s 
7. 7 / -2  S 


309.  Rule. — W?  'He  the  numbers  so  that  the  decimal  points  are  in  the  same  col- 
umn and  proceed  as  in  simple  numbers. 

Note. — If  there  are  fewer  places  in  the  minuend  than  in  the  subtrahend,  consider  the  vacant 
places  in  the  minuend  as  being  ciphers. 


WRITTEN  EXAMPLES 

310.  1.  From  twelve  and  fifty-six  thousandths,  take  three  hundred  forty- 
seven  ten-thousandths. 

2.  Find  the  difference  between  13.8657  and  9.8769. 

3.  Find  the  difference  between  71.8654  and  .587. 

V Find  the  difference  between  10.0101  and  9.00999. 

5.  Find  the  difference  beeween  $371.64  and  $78,725. 

6.  Find  the  difference  between  $2875.625  and  $389.15. 

7.  From  46.201  take  17.9624. 

8.  From  one  take  ninety-nine  hundredths. 

9.  From  .016  take  .0016. 

10.  Subtract  101.027  from  206.431. 

11.  Subtract  39.416712  from  100.01. 

19.  What  is  the  difference  between  209.41  and  8.73546? 

13.  102000.102—102.00102=? 

IV  19.04039 — 6.52781=how  many? 

15.  From  twenty-five  hundred  and  ninety-nine  thousandths,  take  five  hun- 
dred ninety-nine  thousandths. 

16.  Find  the  difference  between  111.3S056  and  22.0605942. 

17.  Find  the  difference  between  160  and  101.00101. 

18.  Find  the  difference  between  36.3636  and  .304403. 

19.  Find  the  difference  between  100.001  and  10100101. 

50.  From  19.413^-  take  lS.45f  and  add  17.06J  to  result. 

51.  Subtract  4.05  -g3^-  from  16.812^-. 

55.  89.723^1— 17.09=hr+250.007R 

S3.  From  72.62f  take  13.614^  and  add  37-g11y  to  result. 

SV  3080.015— 2582.36fL+ 1201 TV 
25.  Subtract  92.08^  from  150.3-g5T. 

56.  From  52.008T5¥  take  2.08-g-3Y. 

27.  From  10.01^  take  (.1001  #+ 8.01  £). 


DECIMALS 


81 


MULTIPLICATION  OF  DECIMALS 

311.  How  to  multiply  decimals. 


250. 


(i) 

.3  f 

2 7 3 
/ S 

.0  / 3 3~ 


4 7 S.  2 S 

,3./  £_ 

3 r o s o o o 

y 7 6 6>  2 s 

/ a z t,  r ? 

/ s / 2 i/  r 7 s o 


70  ' 


by  3.18  ; (3)  $4.8665  by 

(3) 

a)  u.ir  & & 

/ 2 / 

2 S 

/ 2 / 

312.  Rule. — Multiply  as  in  whole  numbers  and  point  off  as  many  decimal  places 
in  the  product  as  there  are  decimal  places  in  both  multiplicand  and  multiplier. 

Note.  — The  small  figures  to  the  right  in  Examples  1 and  2 above  are  proof-figures,  the  first  being 
proved  by  casting  out  the  9’s,  and  the  second  by  casting  out  the  ll’s. 


WRITTEN  EXERCISE 

313.  1.  Multiply  278.05  by  231. 

2.  Multiply  2150.42  by  868. 

3.  Multiply  28.575  by  3.1416. 

Ip.  Multiply  33.5  by  33.5  by  .7854. 

5 Find  the  product  of  3. 5X3.5 X. 5236. 

6.  Multiply  347.176  meters  by  39.37. 

7.  Multiply  875.65  francs  by  .193. 

8.  Multiply  £68.125  by  4.8665. 

9.  Multiply  475.5  liters  by  1.0567. 

10.  Multiply  250.05  hectoliters  by  2.8375. 

11.  Multiply  478.5  kilometers  by  .62137. 

12.  Multiply  6.5  meters  by  5.4  by  2.5  by  1.375. 

13.  Multiply  175.6  kilos  by  .18. 

Iff  The  diameter  of  a circle  is  14.7  feet ; to  find  its  circumference  multiply 
the  diameter  by  3.1416.  What  is  the  circumference? 

15.  Multiply  5.5  by  5.5  and  that  product  by  .7854. 

16.  Multiply  $1575.25  by  .06  and  that  product  by  287.17. 

17.  Multiply  5050.65  by  .01125. 

18.  10.89X108.9X1.089=? 

19  33  X 33  X 33 X .5236=  ? 

20.  25.5 X 25.5 X. 7954=  ? 

21.  What  is  the  product  of  375f  and  101.29? 

22.  What  is  the  value  of  312.2  meters  of  silk  at  5.18  francs  a meter? 

23.  Multiply  15.25  liters  by  7.55  and  that  product  by  .193. 

2ff  A ship  having  a cargo  valued  at  $22500  being  disabled  at  sea,  .22}  ot  its 
cargo  was  thrown  overboard ; what  was  the  loss  to  a shipper  who  owned  .18  of 
the  cargo  ? 


82 


DECIMALS 


DIVISION  OF  DECIMALS 

314.  To  divide  by  a whole  number,  a pure  decimal  or  a mixed 
decimal. 


315.  The  following  method  will  be  found  easy  and  safe  for  beginners,  as  it 
avoids  errors  in  pointing  off  the  decimal  places. 


Examples. — (1)  $198.72=9  ; (2) 
(3)  .409652-^-47  ; (4)  231.3937=78,54. 


.75=. 625 ; 


J / # r. 

7 7 

2 2 

p r 

3 0 

./o  2 f . y S'  <? 

' / r 7 


2 vl.o? 


2 

^7 

2/ 

a 

7 

/ + 

>2  3 

/ . 

3 

f 

7 

3 

7 

0 

0"' 

/ vT 

,7 

0 

r 

7 

+ 

3 

/ 

3 

7 

<2 

7 

r 

If 

3 

7 

2 

7 

7 

3 

/ 

V 

7 

+ 

r 

3 

/ 

0 

2/ 

7 

/ 

2 

2/ 

/ 

2+  7 

7 

3 

7 

7 

v+ 

<7 

7 

0 

0 

o o f y / &> 

*£  <7  y 7 cT  .2 

3 7 & 

3 3 & 

3 2 f 
7 f 
-4-7- 

2 f 2 


2.  r z 

Result  .008716. 


Result  2.9461  -(-. 
Proof 

Divisor  Rem.  6. 
Quotient  Rem.  4 ; 

6X4=24=6. 
Dividend  Rem.  1. 
Remainder,  4. 
(1+9)— 4=6. 


316.  Rule. — Write  the  divisor  on  the  left  of  the  dividend  with  a curved  line 
between  them,  as  in  the  division  of  whole  numbers. 

Draw  a horizontal  line  below  the  dividend  if  the  division  is  to  be  “ short,”  or 
above  it  if  the  question  requires  long  division. 

When  the  divisor  contains  more  decimal  places  than  the  dividend,  equalize  them 
by  annexing  ciphers  to  the  dividend.  ( See  Ex.  2) 

Draw  a vertical  line  through  the  dividend  (at  right  angles  to  the  horizontal  line) 
as  many  places  to  the  right  of  the  decimal  point  as  there  are  decimal  places  in  the 
divisor — through  the  decimal  point  when  the  divisor  is  a whole  number. 

Proceed  with  the  division,  being  careful  to  write  the  first  figure  of  the  quotient 
directly  beneath  (in  short  division ) or  directly  above  (in  long  division)  the  right-hand 
figure  of  the  first  partial  dividend.  All  that  part  of  the  quotient  to  the  left  of  the  deci- 
mal line  will  be  a whole  number,  and  that  to  the  right  of  it  a decimal. 


DECIMALS 


83 


WRITTEN  EXERCISE  1 
317.  Obtain  exact  results  for  the  following  : 


1. 

Divide 

.625  by  2.5. 

2. 

Divide 

15.25  by  .05. 

3. 

Divide 

.0156  by  .003. 

4- 

Divide 

23  1 by  .07. 

5. 

Divide 

2.31  by  .7. 

6. 

Divide 

345.15  by  .75. 

7. 

Divide 

7 5 by  .015 

8. 

Divide 

37.06  by  .017. 

9. 

Divide 

142.74  by  .61. 

10. 

Divide 

110  by  .44. 

11. 

Divide 

24.75  by  2.25. 

12. 

Divide 

3.65  by  7.3. 

13. 

Divide 

2425.5  by  .32. 

14- 

Divide 

56.43  by  .064. 

15. 

Divide 

79.64  by  5.5. 

16. 

Divide 

30.25  by  16 

17. 

Divide 

2150.4  by  67.2. 

18. 

Divide 

5280  by  16  5. 

19. 

Divide 

43560  by  .375. 

20. 

Divide 

.7854  by  .0034. 

21. 

Divide 

9 5 byl9. 

22. 

Divide 

36  5 by  73. 

23. 

Divide 

1750  by  .875. 

24. 

Divide 

461.975  by  5.435. 

25. 

Divide 

3.6  by  .18. 

26. 

Divide 

343  68  by  53.7. 

27. 

Divide 

44.44  by  1.1 . 

28. 

Divide 

128.625  by  .08. 

29. 

Divide 

169.39  by  13. 

30. 

Divide 

162.54  by  27. 

31. 

Divide 

5236  by  .008. 

32. 

Divide 

7276  5 by  31.5. 

33. 

Divide 

.07958  by  .16. 

34. 

Divide 

93.75  by  .0625. 

35. 

Divide 

272.25  by  25. 

36. 

Divide 

39.37  by  .32. 

37. 

Divide 

30.25  by  6.4. 

38. 

Divide 

165  by  .055. 

39. 

Divide 

183.15  by  55. 

40. 

Divide 

105.67  by  3.2. 

WRITTEN  EXERCISE  2 


318.  Extend  the  following  to  at  least  four  decimal  places  when  necessary  : 


1.  Divide  .175  by  2. 

2.  Divide  35.104  by  5. 

3.  Divide  .3562  by  5. 

4-  Divide  360  by  .06. 

5.  Divide  14.2672  by  .08. 

6.  Divide  360  by  .04J. 

7.  Divide  568.6  by  30.25. 

8.  Divide  765.28  by  2.3J. 

9.  Divide  360  by  .05. 

10.  Divide  4.8665  by  .01}. 

11.  Divide  2150.4  by  8. 

12.  Divide  48.4  by  .16. 


13.  Divide  24  255  by  .11. 
14..  Divide  384  by  1.2. 

15.  Divide  3.45  by  .005. 

16  Divide  30.4  by  19. 

17.  Divide  16.9  by  .13. 

18.  Divide  37.5  by  .00}. 

19.  Divide  7276.5  by  31.5. 

20.  Divide  864.65  by  272}. 

21.  Divide  375  bj1 2 3 * 5 6 7 8 9 10 11 12  .19}. 

22.  Divide  56}  bv  .13}. 

23.  Divide  272}  by  .375. 

24-  Divide  34.33}  by  .03}. 

25.  Divide  17.16§  by  83}. 


Note. — A repeating  decimal  may  be  extended  to  the  required  number  of  places  in  the  dividend, 
but  should  be  handled  as  a mixed  number  in  the  divisor. 


84 


DECIMALS 


MISCELLANEOUS  EXERCISE  3 

319.  Ca  rry  results  to  four  decimal  places  when  necessary: 


1.  Divide  4378.75  by  320. 

2.  Divide  9654.5  by  24.75. 

3.  Divide  138.48  by  30±. 

4-  Divide  7500  by  24f. 

5.  Divide  2747.7  by  57.75. 

6 Divide  39.37  by  36. 

7.  Divide  5.18  by  .193. 

8.  Divide  4.8665  by  240. 

9.  Divide  .7854  by  231. 

10.  Divide  329.84  by  5.19. 

11.  Divide  82432  by  .37. 

IS.  Divide  12.4328  by  .61. 

13.  Divide  $245.50  by  .23. 

14-  Divide  $545.25  by  12.3. 

15.  Divide  $1250.60  by  $1  25. 

16.  Divide  34.5685  by  .97. 

17.  Divide  112.3456  by  1.37. 


18.  Divide  3.39924  by8.716. 

19.  Divide  .00540625  by  .00865 

50.  Divide  21.984375  by  78.65. 

51.  Divide  21.984375  by  .375. 
SS.  Divide  $86.73  by  1239. 

S3.  Divide  $939.12  by  13416. 

#4-  Divide  $13878.72  by  $48.19. 
S5.  Divide  $9615.36  by  $50  08. 

26.  Divide  .409652  by  .047. 

27.  Divide  3 1.833 J by  18-f-. 

28.  Divide  96.8  by  .238. 

29.  Divide  175.25  by  .000f. 

30.  Divide  62.8  bv  .14. 

31.  Divide  653.21056  by  1.8 

32.  Divide  94.265  by  32. 

33.  Divide  242.55  by  75.796J. 
54-  Divide  741.85f  by  124. 


APPLICATION  OF  DECIMALS  TO  INVENTORIES 

320.  It  is  usually  sufficient  in  ordinary  commercial  operations  to  carry  a 
decimal  division  to  four  places,  but  in  estimating  the  cost  of  manufactured  prod- 
ucts to  get  a unit  of  value  on  millions,  it  is  necessary  to  carry  the  result  to  eight 
or  nine  decimal  places.  This  is  true  also  in  figuring  bankruptcy  and  part- 
nership settlements  by  the  percentage  method,  and  in  constructing  compound 
interest  tables. 


WRITTEN  PROBLEMS 

321.  1.  If  the  cost  of  producing  7500000  barrels  of  a certain  grade  of  flour 
was  $31829437.95,  what  should  be  the  company’s  inventory  value  of  2345560 
barrels  ? 

2.  The  cost  of  producing  5000000  tons  of  pig  iron  is  $47953274.87,  what 
should  be  the  mill  inventory  value  of  2897450  tons? 

3.  If  the  cost  of  manufacturing  7586000  rolls  of  building  paper  was 
$4492673.65,  what  should  be  the  manufacturer’s  inventory  of  3815967  rolls? 

4-  The  cost  of  producing  9847600  rolls  of  wall  paper  was  $102S922.45,  what 
should  be  the  manufacturer’s  inventory  value  of  31472569  rolls? 

5.  If  the  cost  of  making  44562975  pounds  of  a certain  commodity  was 
$2943647.50,  what  should  be  the  manufacturer’s  inventory  value  of  29675875 
Dounds  ? 

x 

6.  If  the  cost  of  refining  24756800  gallons  of  oil  was  $12S5675.70,  what  should 
be  the  company’s  inventory  value  of  6938450  gallons? 

7.  The  cost  of  making  25000000  tons  of  steel  was  $487755692,  what  was  the 
company’s  inventory  value  of  9398670  tons? 


DECIMALS 


85 


SHORT  METHODS  IN  DECIMALS 

Multiplication 

322.  To  multiply  by  simply  moving  the  decimal  point. 

1.  Multiply  24.75  by  10.  f.  Multiply  .5236  by  100. 

2.  Multiply  31.5  by  100.  5.  Multiply  2150.4  by  10000. 

3.  Multiply  3.1416  by  1000.  6.  Multiply  172.8  by  10. 

323.  Rule. — Moving  the  decimal  point  one  place  to  the  right  multiplies  by  10 ; 

two  places , by  100  ; three  places , by  1000,  etc. 

324.  To  multiply  by  a decimal,  the  equivalent  of  which  is 
etc.,  of  1. 


1 

2) 


1.  Multiply  43560  by  .5. 

2.  Multiply  5280  by  .25. 

3.  Multiply  2150.4  by  .125. 
4-  .Multiply  2747.7  by  .33J. 
5.  Multiply  24.75  by  .16|. 


6.  Multiply  31.5  by  .081 


7.  Multiply  4.8665  by  12J. 

8.  Multiply  231  by  .14-f-. 

9.  Multiply  160  by  .2J. 

10.  Multiply  78.54  by  .3334- 

325.  Rule. — Divide  by  the  denominator  of  the  equivalent  common  fraction. 

326.  To  multiply  by  a decimal,  the  equivalent  fraction  of  which  is 
l,  etc.,  greater  or  less  than  1. 


1.  Multiply  2150.4  bv  .75. 

2.  Multiply  2747.7  by  .66 1. 
3:  Multiply  31.5  by  .83^-. 

4-  Multiply  5280  by  874. 

5.  Multiply  4.86  by  .99. 

6.  Multiply  43560  by  .91f. 


7.  Multiply  2150  4 by  1.25. 

8.  Multiply  2747.7  by  1.334. 

9.  Multiply  31.5  by  1.16f. 

10.  Multiply  5280  by  1.125. 

11.  Multiply  4.8665  by  1.10. 

12.  Multiply  43560  by  I.O84. 


327.  Rule. — From  the  multiplicand  subtract  such  part  of  itself  as  the  multiplier 
is  less  than  l,  or  add  such  part  as  it  is  greater  than  1. 


MISCELLANEOUS  EXERCISE 


328.  Multiply 

1.  30.25x2.5. 

2.  3.1416X25. 

3.  31.5X12.5. 

I 24.75  X 33J-. 

5.  4.8665x500. 


6.  30.25x74- 

7.  3.1416X750. 

8.  24.75X874. 

9.  2150.4  X6f. 
10.  31.5X66§. 


11.  30.25X125. 

12.  3.1416X1250. 

13.  .238X112.5. 

If.  2150.41  X 1334. 
15.  .193X1500. 


Division 

329.  To  divide  by  moving  the  decimal  point. 

1.  Divide  24.75  by  10.  5.  Divide  37.85  by  100. 

2.  Divide  2150.4  by  100.  6.  Divide  4.8665  by  1000. 

3.  Divide  31.5  by  1000.  7.  Divide  31.416  by  10. 

f.  Divide  1728  by  100.  8.  Divide  144  by  100. 

330.  Rule. — Moving  the  decimal  point  one  place  to  the  left  divides  by  10  ; two 
places,  by  100  ; three  places,  by  1000,  etc. 


86 


DECIMALS 


331.  To  divide  by  a decimal  of  which  the  equivalent  fraction  is 
J,  etc. 


1.  Divide  3.1416  by  .5. 

2.  Divide  .7854  by  .25. 

3.  Divide  31.5  by  .334. 
I/..  Divide  24.75  by  .16f. 

5.  Divide  16  5 by  .12J. 


6.  Divide  144  by  .08J. 

7.  Divide  1728  by  ,14-f-. 

8.  Divide  2150.4  by  .125. 

9.  Divide  4.8665  by  ,2J. 

10.  Divide  30.25  by  .3334 


332.  Rule — Multiply  by  the  denominator  of  the  equivalent  fraction. 


333.  To  divide  by  a decimal  whose  equivalent  value  is  4,  4>  etc., 
greater  or  less  than  1. 


Example— 36  a-. 75=36  a-|=36X  3=48. 

Suggestion. — 36X4  is  equivalent  to  adding  4 of  the  dividend  to  itself. 


1.  Divide  24  75  by  .75. 

2.  Divide  2150.4  by  .874. 

3.  Divide  2747.7  by  .66|. 
f.  Divide  144  by  .834. 

5.  Divide  1728  b}r  ,85-f-. 

6.  Divide  31.5  by  .74  ; by  ,6f. 


7.  Divide  24.75  by  1.25. 

8.  Divide  2150.4  by  1.334 

9.  Divide  2747.7  by  1.50. 

10.  Divide  144  by  1.124. 

11.  Divide  1728  by  1.25. 

12.  Divide  31.5  by  1.16f. 


334.  Rule. — Divide  by  the  numerator  of  the  equivalent  common  fraction  and 
add  the  result  when  the  divisor  is  less  than  2,  or  subtract  when  it  is  greater  than  1. 


335.  To  divide  by  a whole  number  ending  in  ciphers. 


1.  Divide  435.6  by  300. 

2.  Divide  528.85  by  700. 

3.  Divide  75864.15  by  6000. 
f Divide  4576.75  by  4000. 


5.  Divide  21504  by  320. 

6.  Divide  9178.39  by  1300. 

7.  Divide  37477.88  by  11000. 

8.  Divide  4866  5 by  1600. 


336.  R ule. — Drop  the  ciphers  on  the  right  of  the  divisor , move  the  decimal 
point  in  the  dividend  an  equal  number  of  places  to  the  left , and  divide. 


REVIEW  EXERCISE 

337.  In  simplifying  expressions  connected  by  the  signs  of  the  fundamental 
operations  ( + , — , X,  and  a-)  observe  the  following  order  : 

1.  Multiplication. 

2.  Division. 

3.  Addition  and  Subtraction  as  they  may  occur. 

Expressions  contained  in  marks  of  aggregation,  (parenthesis,  brackets, 
braces)  must  be  simplified  according  to  the  above  order,  and  the  result  used  for 
the  whole  expression  in  the  marks;  then  simplify  the  original  expression  as 
directed  above. 


DECIMALS 


Example.— 4X8+14=8+  [16— (25=5)  +7]=29. 
Operations : (Restatement). 

1.  8+14=22.  a.  4X22=88. 

2.  (25=5)=5.  b.  88=8=11. 

3.  (16— 5+7)=18.  c.  11  + 18=29. 

+ 4x22=8+18 


338.  1-  12.5  XI  2.5  X. 7854=144=? 

2.  26.728  X $1.05=  ? 

3.  13.450  X $18.25=? 

+ 43.75X$11  + 132.80X$7=  ? 

5.  184.80  X $.625=? 

6.  1 2.456  X $8.75=? 

7.  28165X$.08=$28.75=  ? 

8.  £765  5 X $4.8665=  ? 

9.  £12345  6 X $4.8665=$. 193=? 


10.  687.5X39.37=36=? 

11.  785.125X36=39.37=? 

12.  7912X25000X640=? 


13.  791 2 X 791 2 X 791 2 X. 5236=? 


U 


24X20X8X4  x$025=s? 


15. 


60X24X30  9 

144 


16.  60x49xl44=(14x20)=  ? 

17.  17X22X62.5=? 

18.  (28  X 6600=9)  X $.92=  ? 

19.  ($900=.05)X  1.1675=  ? 

20.  ($9.21=. 03) X. 65=  ? 


0)1  75x5280x48  *-O0_ 

2L  43660  + *'2U- 


22.  60X3.1416X8X1728=231=? 

23.  (1 ,5  X 5280  X 75=272.25)  X $1 .10 
21  187X187  X. 7854=272.25=? 

25.  2240X7=8X4X168=? 

26.  14X3.1416X38=9=? 

27.  59X5280=60=5.5=? 

28.  $76000 X. 75X1.04=? 

$950 X 93 X. 06  9 

360  ~ • 

Yl  $6500  X 61  X .05 9 

"365  ~~  ' 

$1896  X (91 — 8)  X .06  9 

360  — ' 

32.  £124.5025  X $4.87+ $1.50=  ? 

33.  Divide  7.5  by  62.5. 

3£.  $1890=.045  X 1.05=  ? 

35.  $1940.50=4.875=? 

36.  640 X $6.25  X. 02=  ? 

37.  ($7200=.08)X  1.1625=? 

38.  £37.804  X $4.8665=  ? 

39.  4775 X $.82— (4175 X $-82 X. 04)= 
4.0.  (2X40  + 2 X60)X9X 3.5 X $.30= 

41.  $972.50=(162X 4=9 X $3.10)= 

42.  15290  bu.=92  bu.=102=  ? 


43.  165+3. u75 — .60863x40.07=.  1309=  ? 

44-  (2X16X9)+(2X18X9)+(16X18)=? 

4.5.  102  X 102  X 102  X .5236  X 1728=231=  ? 

46.  (22X22 X .7854)= 144 X 24 X 450 X $16.50=  ? 

47.  2240  X 12=(26  X 26  X .7854=144  X 1 66)=  ? 

48.  144  X 144  X. 7854  X8Xl728=(31. 5X9X24)=  ? 

49.  $31 7829.32+ $61378.12 — ($46312.85 +$301449.72)=  ? 

50.  (385  X 385  X .7854=43560)  X $1.59=  ? 


88 


DECIMALS 


REVIEW  PROBLEMS 


339.  1.  $197.87*  + $67.37f-K$27.47ix3|)— .75^=? 
9.  $37i+$1.87f+$g; 


4 


3.  Find  the  cost  of  10  loads  of  hay  at  $16J  per  ton,  containing  respectively 
2|  tons,  tons,  1.375  tons,  2\  tons,  2.8  tons,  1-^  tons,  If  tons,  2.15  tons,  If  tons 
and  24  tons. 

f.  If  9.3  barrels  of  flour  cost  $48,825.  what  will  15.25  barrels  cost  at  the 
same  rate  ? 

5.  A farmer  sowed  three  fields  in  wheat,  containing  respectively  1171- 
acres,  125§  acres,  and  325.561  acres.  He  harvested  from  the  first  an  average  of 
27J  bushels  per  acre,  from  the  second  30.25  bushels  per  acre,  and  from  the  third 
29f  bushels  per  acre.  Find  the  value  of  his  wheat  at  $1.12J  per  bushel. 

6.  A speculator  bought  325.45  acres  of  land  at  $15f  per  acre  and  sold  it  in 
plots  of  1.15  acres  each  at  $112.50  a plot.  Find  his  gain  or  loss. 

7.  Two  trains  are  587.25  miles  apart,  and  are  running  toward  each  other. 
One  train  has  a speed  of  45f  miles  per  hour  and  the  other  53J  miles  per  hour. 
After  they  have  been  running  for  4.6  hours,  how  far  are  they  apart? 

8.  A grain  dealer  bought  1275f-  bushels  of  wheat  at  $.874  per  bushel.  He 
sold  f of  it  at  $.90f  per  bushel,  .25  of  the  remainder  at  $.94f  per  bushel,  and 
what  still  remained  at  cost.  What  did  he  gain? 

9.  If  a bushel  contains  2150.4  cubic  inches,  how  many  bushels  in  37560 
cubic  inches?  What  is  the  value  of  this  quantity  at  624  cents  per  bushel? 

10.  During  a year  a man  spends  .37J  of  his  income  for  clothes,  .274  for 
board,  .16  for  incidentals,  and  deposits  the  rest  in  bank.  If  his  bank  deposit  is 
$570,  what  is  his  income? 

11.  A merchant  deposited  in  a bank  at  one  time  $575.25,  and  at  another 
$487.30.  If  he  checked  out  $387.50,  how  many  yards  of  cloth  can  he  buy,  at 
$1.25  per  yard,  with  the  money  he  still  has  in  bank? 


13.  If  .625  of  a ton  of  coal  cost  $3.25,  what  will  7f  tons  cost  ? 

A£.  Divide  $9  between  two  boys  so  that  one  shall  receive  .25  more  than  the 
other. 

15.  If  a load  of  hay  weighing  1665  pounds  cost  $8.75,  how  much  should,  at 
that  rate,  be  paid  for  a long  ton  (2240  pounds)? 

16.  How  many  barrels  of  flour  at  $5,124  must  be  given  for  127.374  bushels 
of  com  at  30  cents  per  bushel  and  127J  bushels  of  wheat  at  $.874  per  bushel  ? 

17.  If  the  freight  on  75.8  barrels  of  apples  is  $9.47^,  what  should  be  paid 
on  103f  barrels  ? 

18.  If  there  are  63360  inches  in  a mile,  and  39.37  inches  in  a meter,  what 
will  it  cost  to  pave  a street  one  mile  long  at  $1.12J  per  meter? 

19.  The  circumference  of  a body  is  3.1416  times  the  diameter.  What  is  the 
diameter  of  the  earth  if  the  circumference  is  24899.024  miles  ? 


34 

12.  Change to  a decimal  fraction. 


3 


DECIMALS 


89 


20.  The  value  of  a franc  in  United  States  money  is  $.193.  How  many  francs 
in  $458.37 J,  and  how  many  yards  of  silk  could  be  purchased  with  this  sum  at 
9.5  francs  per  yard  ? 

21.  If  .37  of  a boy’s  wages  for  a week  is  $3.14J,  what  should  he  receive  for 
7f  weeks  ? 

22.  If  a certain  kind  of  iron  ore  contains  .37J  foreign  substance,  how  much 
pure  iron  in  968.9  tons  of  the  ore  ? 

23.  A tailor  bought  32  pieces  of  cloth  containing  23J  yards  each  at  $3.60J 
per  yard  and  sold  it  so  as  to  gain  $372,  after  allowing  $14.88  for  freight.  How 
much  did  he  receive  per  yard? 

24..  A man  in  disposing  of  his  property  gave  J to  his  younger  son,  J of  the 
remainder  to  his  elder  son,  .125  to  a church,  and  the  remainder,  $32151.35,  to  his 
wife.  Find  value  of  his  property  and  amount  each  son  received. 

25.  A clerk  spent  .37J  of  his  money  and  still  had  remaining  $237.25.  How 
much  did  he  spend? 

26.  A house  and  lot  together  cost  $7805.38.  If  the  house  cost  1.75  times  as 
much  as  the  lot,  what  was  the  cost  of  each  ? 

27.  The  product  of  three  numbers  is  331.81071,  and  two  of  the  numbers  are 
3.79  and  23.1.  What  is  the  third  number? 

28.  A grocer  invested  $526.50  in  Rio  and  Java  coffee.  If  .625  of  the  amount 
invested  in  Rio  equals  the  amount  invested  in  Java,  what  sum  was  invested  in 
each,  and  how  many  pounds  of  each  could  he  buy  at  22|-  cents  per  pound  ? 

29.  At  the  end  of  the  first  year  a merchant’s  inventory  is  $37325. 62J,  which 
is  equal  to  .375  of  his  sales  for  the  year.  If  during  the  year  he  gained  $12725. 37J, 
what  was  the  amount  of  his  purchases  ? 

30.  Two  men  commenced  business  with  equal  capital.  At  the  end  of  the 
year  one  had  gained  $3725.40,  the  other  had  lost  $2376.80,  and  together  they  had 
$7897.30.  How  much  had  each  at  first  ? 

31.  A merchant  sold  450  yards  of  cloth  from  a piece  containing  875  yards. 
What  decimal  part  of  the  piece  remained  unsold  ? 

32.  A,  B and  C own  a mill.  A owns  .265  of  it,  B .378  of  it,  and  C the 
remainder.  If  A’s  share  is  $7234.50,  what  is  the  value  of  B’s  and  C’s  shares? 

33.  If  .375  of  a ton  of  coal  cost  $4.95,  what  will  .0875  of  a ton  cost? 

34-  A invested  .375  of  his  money  in  a factory,  .24  of  the  remainder  in  a city 
property,  and  deposited  the  balance,  $17432.50,  in  a bank.  Find  amount  of  his 
money,  and  amount  invested  in  the  factory. 

35.  The  private  secretary  of  the  governor  receives  $2000,  of  which  he  spends 
$525  for  rent,  $675  for  living  expenses,  $297.50  for  clothes.  What  decimal  part 
of  his  salary  does  he  save? 

36.  If  25.5  barrels  of  flour  cost  $85,  how  many  can  you  get  for  $40.25  ? 

37.  A man  traveled  697^  miles  by  rail  and  011  bicycle.  If  he  traveled  .55  as 
far  on  bicycle  as  by  rail,  how  far  did  he  travel  by  rail  and  how  far  on  bicycle? 

38.  If  the  value  of  an  English  sovereign  is  $4.8665,  how  many  sovereigns 
are  worth  $100  ? What  is  the  value  of  156  sovereigns  ? 


QUANTITY,  PRICE  AND  COST 


340.  The  relation  of  these  three  elements  is  that  of  multiplicand,  multi- 
plier and  product. 


Quantity  is  the  number  of  units. 

Price  is  the  cost  of  one  unit. 

Cost  is  the  value  of  a quantity. 

Formula : 

1.  Quantity  X Price=Cost. 

2.  CostH-Price=Quantity. 

3.  Cost-uQuantity=Price. 


341.  An  aliquot  is  a number  or  quantity  (either  integral  or  fractional)  that 
is  contained  in  another  number  or  quantity  an  exact  number  of  times.  Thus,  6 
is  an  aliquot  of  18;  25  is  an  aliquot  of  100 ; J is  an  aliquot  of  4,  and  of  f. 

342.  An  aliquot  part  of  a number  or  quantity  is  one  of  its  even  parts,  which 
is  given  a fractional  expression  by  placing  it  over  such  number  or  quantity  and 
reducing  it  to  its  lowest  terms. 

343.  A simple  aliquot  is  one  that  can  be  reduced  to  a fractional  expression 
having  a numerator  of  1 ; as  i,  4,  etc. 

344.  A compound  aliquot  is  comprised  of  two  or  more  simple  aliquots ; as, 

t=f+(i  ofi). 

345.  An  aliquot  unit  is  the  number  or  quantity  of  which  the  aliquot  is  an 
even  part. 


346.  A commercial  unit  is  a fixed  unit  of  measure  upon  which  the  price 
of  a commodity  is  based  ; as,  a yard,  a bushel,  a gallon,  a pound,  a ton,  etc. 


347. 

Simple 

Aliquot 

Parts  of 

$1.00. 

50c. 

1 

2 

16fc. 

1 

6 

10c. 

— i 

1 0 

6|c 

— i 
i r> 

33g-c. 

— 1 
3 

14fc. 

1 

7 

9QrC. 

— i 
i i 

5c. 

- i 
2 0 

25c. 

1 

4 

124c. 

— 1 
8 

Sic. 

- i 
1 2 

3|c 

— i 
• — To 

20c. 

— 1 
5 

H|c. 

1 

9 

6fc. 

- 1 
1 5 

24c, 

- i 

■ 40 

348. 

If  the 

price  given  should  be  $1,334,  or  4 

more  than 

$1.00;  or 

66§  c 

i less  than  $1.00,  the  result  can  be  readily  obtained  by  adding  or  subtracting  the 
fractional  part,  as  the  case  may  be.  This  can  be  applied  to  every  simple  aliquot 
with  any  easy  unit  as  a base,  whether  25c.,  50c.,  $1.00,  $100  or  100  per  cent..,  the 
object  being  to  simplify  multiplication  and  division  in  business  computations. 

Require  students  to  make  up  tables  of  the  aliquot  parts  of  50c.,  25c.,  10c., 
similar  to  table  given  above. 


90 


QUANTITY,  PRICE  AND  COST 


91 


APPLICATION  OF  ALIQUOT  PARTS 

349.  General  Rule. — Divide  the  quantity  by  the  number  of  commercial  units 
that  $ 1 will  buy. 

Explanation — At  121c.,  $1  will  buy  8 units  ; at  33Jc.,  3 units  ; at  50c.,  2 
units;  at  66|c.,  f units,  or  11;  at  75c.,  f or  1£  units;  at  $1.25,  4 of  a unit ; at 
$1.33J,  | of  a unit ; at  $1.50,  § of  a unit. 

Note. — -When  the  quantity  contains  a common  fraction,  it  should  he  reduced  to  a decimal  before 
dividing.  If  the  fraction  will  not  reduce  to  a simple  decimal,  extend  it  to  4 decimal  places. 

350.  To  find  the  cost  when  the  price  is  a simple  aliquot  of  $1. 


Find  the  cost  of 


Exercise  1 


1.  175  yds.  @ 50c. 

2.  84f  yds.  @ 33Jc. 

3.  127  gal.  @ 25c. 
If.  75J  gal.  @ 20c. 

5.  156  lbs.  @ 16fc. 


6.  99tL  yds.  @ 12Jc. 

7.  350  yds.  @ 14fc. 

8.  894  lbs.  @ 10c. 

9.  810  qts.  @ 114c. 

10.  495J  qts.  @ 9^0. 


11.  480  gal.  @ 8Jc. 

12.  230  yds.  @ 6jC. 

13.  2791  yds.  @ 331c. 

i.£.  236f  lbs.  @ 25c. 
15.  7S9f-  yds.  @ 16fc. 


351.  To  find  the  cost  when  the  price  is  a simple  aliquot  less  than  $1 
or  more  than  $1. 


Exercise  2 


1.  248  gal.  @ 871c. 
2 3561  gal.  @ 90c. 
3.  189  yds.  @ 75c. 
If.  456  yds.  @ 66§c. 

5.  360f  yds.  @ 80c. 

6.  35 6 gal.  @ 95c. 


7.  750  yds.  @ 871c. 

8.  632  yds.  @ $1.50. 

9.  156|  yds.  @ $1.33^. 

10.  256  doz.  @ $1.25. 

11.  1691  doz.  @ $1.16f. 

12.  573|  yds.  @ 75c. 


352.  Rule. — Add  to  the  quantity  or  subtract  from  it  such  aliquot  part  of  itself 
as  the  price  is  greater  or  less  than  $1. 


353.  To  find  the  cost  when  the  price  is  a simple  aliquot  more  or  less 
than  $2,  $3,  $4,  $5,  etc. 

Exercise  3 


1.  960  bbls.  @ $2,121. 

2.  480  bbls.  @ $2.16|. 

3.  560  bbls.  @ $3.14|-. 
If..  728  bbls.  @ $3.25. 

5.  360  bbls.  @ $4,331. 


6.  244  yds.  @ $1.87f. 

7.  342  yds.  @ $1.83f 

8.  572  yds.  (cy  $1.75. 

9.  456f  yds.  @ $2,331 

10.  196  yds.  @ $2.75. 


11.  128  tons  @ $5.12J. 

12.  96f  tons  @ $4.66f. 

13.  240  tons  @ $3.87J. 
If  85J  tons  @ $6.25. 
15.  375  tons  @ $2.66f. 


354.  R ule. — Multiply  the  quantity  by  the  nearest  dollar  price  and  add  to  the 
result,  or  subtract  from  it,  such  aliquot  part  of  the  quantity  as  the  real  price  is  above  or 
below  the  assumed  price. 


92 


QUANTITY,  PRICE  AND  COST 


355.  To  find  the  cost  using  50c.,  25c.,  or  10c.  as  a base. 


Find  the  cost  of 

1.  164  yds.  @ 55c. 

2.  132  gal.  @ 45c. 

3.  284  gal.  (aj  374c. 
Ip.  716  gal.  @ 56|c. 


Exercise  4 

5.  76J  yds.  @ 224c. 

6.  120  yds.  @ 18fc. 

7.  92J  yds.  @ 35c. 

8.  132  yds.  @ 15c. 


9.  48f  yds.  @ 134c, 

10.  152  lbs.  @ 7Jc. 

11.  69f  lbs.  @ 6fc. 

12.  175  lbs.  @ 3Jc. 


Note.— These  exercises  are  to  test  the  student’s  power  of  invention. 

356.  To  find  the  cost  when  the  price  is  an  aliquot  part  of  $10,  $100, 
$1000,  or  an  aliquot  part  more  or  less  than  $10,  etc. 


1.  318f  yds.  @ $2.50. 

2.  2744  yds.  @ $3,334- 

3.  157f  yds.  @ $5. 

Ip.  894  tons  @ $334- 


Exercise  5 

5.  534  doz.  @ $15. 

6.  28  bbls.  @ $7.50. 

7.  67J  tons  @ $25. 

8.  39f  tons  @ $33J. 


P.  160  acres  @ $125. 

10.  714  acres  @ $250. 

11.  27f  acres  @ $500. 
7^.  145|  acres  @ $150. 


357.  To  find  the  cost  when  the  quantity  is  10, 100, 1000,  etc.,  or  when 
it  may  be  factored  upon  some  power  of  10  taken  as  an  aliquot  unit. 


1.  10  yds.  @ $1,354. 

2.  100  yds.  @ 73fc. 

3.  1000  yds.  @ 9|c. 
Ip.  500  yds.  @ 86-fc. 


Exercise  6 

5.  250  gals.  @ $1.15|. 

6.  125  gals.  @ $1,194. 

7.  75  lbs.  @ 231-e. 

8.  2500  yds.  @ 64fc. 


9.  1500  gals.  @ 894c. 

10.  2000  lbs.  @ 494c. 

11.  500  tons  @ $3,784-. 

12.  750  bbls.  @ $2.96f 


358.  To  find  the  cost  when  the  numerator  of  the  aliquot  is  greater 
than  1. 


359.  Rule. — Resolve  the  aliquot  into  two  or  more  simple  aliquots. 
Illustration  : 374c.=$f=4  + (i  °f  $)• 


Exercise  7 


1.  144  yds.  @ 374c. 

2.  256  gals.  @ 624c. 

3.  1964  gro-  @ 85c. 

Ip.  288  yds.  @ $1,374- 

5.  512  yds.  @ $1,624 


6.  640  bbl.  @ $1,724. 

7.  84  bbls.  @ $2,374. 

8.  56  bbls.  @ $3,624. 

9.  68  bbls.  @ $2,224. 

10.  96  tons  @ $5,374. 


QUANTITY,  PRICE  AND  COST 


93 


360.  To  find  the  quantity  when  the  cost  is  given  and  the  unit  price 
is  a simple  aliquot  of  $1. 

Exercise  8 


Cost.  Unit  Price.  Cost.  Unit  Price.  Cost.  Unit  Price. 


1. 

$52.50 

25c. 

6. 

$14.85 

3ic. 

11. 

$19.45  20c. 

2. 

368.25 

12ic. 

7. 

8.75 

2ic. 

12. 

73.25  8^c. 

3. 

97.75 

16fc. 

8. 

4.721 

5c. 

13. 

16.50  7ic. 

f- 

142.50 

33JC. 

9. 

184.50 

14f( 

■» 

If. 

24.75  6ic. 

5. 

295.75 

50c. 

10. 

59.10 

11* 

■> 

15. 

36.20  66§c. 

16. 

$78.75  (a 

) 25c.  a 

yard. 

21. 

$175. 

.50  @ $1.25  a yard. 

17. 

$156.25  ( 

@ 33^-c. 

a yard. 

22. 

$264. 

80  @ $1.33J  a yard. 

18. 

$24.75  (a 

} 16fc.  a pound. 

23. 

$363. 

75  @ $1.50  per  barrel. 

19. 

147.25  (a 

) 6Jc.  a 

gallon. 

2f. 

$1975  @ $12.50  a ton. 

20. 

$385  @ 

87£c.  yard. 

25. 

$26.58  @ 75c.  a yard. 

361.  Rui  ,e. — Multiply  the  cost  by  the  number  of  commercial  units  that  $ 1 will 

buy. 

362.  To  find  the  price  of  a commercial  unit. 

Example. — What  is  the  unit  price  when  100  yds.  cost  $43,625  ? 

Solution. — Moving  the  point  in  the  cost  two  places  to  the  left  gives  $.43625,  or  43f  cents,  as  the 
price  of  a yard. 

Exercise  9 


363.  Find  the  unit  price  if 

1.  100  yds.  cost  $15,874. 

2.  1000  lbs.  cost  $66.25. 

3.  250  yds.  cost  $188.75. 
f.  125  yds.  cost  $177.50. 

5.  3000  tons  cost  $116.25. 


6.  333J  yds.  cost  $575. 

7.  2000  lbs.  cost  $127.50. 

8.  150  yds.  cost  $78.75. 

9.  1500  gals,  cost  $97.12J. 

10.  750  yds.  cost  $2986.50. 


364.  Find  the  cost  of 

1.  389  yds.  @ 25c. 

2.  246f  yds.  @ 33Jc. 
3 369  lbs.  @ 16fc. 
f.  166f  yds.  @ 12Jc. 

S.  750  gals.  @ 8Jc. 


Review  Exercise  10 

6.  156  lbs.  @ $1.33j-. 

7.  562J  yds.  @ $2.50. 

8.  116  gals.  @ $1.12J. 

9.  3126  lbs.  @ 2Jc. 

10.  964  ft.  @ 15c. 


11.  1190  bbls.  @ $4.1 2J. 

1 2.  41 8f  yds.  @ $3.16f. 

13.  273J  yds.  @ $3.87|-. 
If.  168  tons  @ $6.25. 
15.  178J  yds.  @ 62Jc. 


94 


QUANTITY,  PRICK  AND  COST 


SPECIAL  RULES 

365.  To  find  the  cost  when  merchandise  is  sold  by  the  hundred  or 
the  thousand. 

Many  kinds  of  merchandise,  such  as  lumber,  cigars,  meal,  grain,  etc.,  are 
sold  by  the  hundred  or  thousand. 

Note.— C is  the  abbreviation  for  100  ; M for  1000.  A cental  of  grain  is  100  pounds.  A cwt. 
is  100  pounds  avoirdupois.  A net  ton  is  2000  pounds  ; a long  ton,  2240  pounds. 

Example. — Find  the  value  of  2604  lbs.  of  wheat  at  $1.12  per  cental. 


366.  Rule. — Point  off  two  places  in  the  quantity  if  the 
price  is  per  hundred,  or  three  places  if  per  thousand  ; multiply 
by  the  price  and  point  off  as  in  the  multiplcation  of  decimals. 


Note. — For  net  tons,  point  off  three  places  and  divide  by  2. 


2 C>.0  4 
/./  2. 
S2  o r 
-2  r 6>  44 
f i f . / £ 4 r 


WRITTEN  PROBLEMS 

367.  1-  Find  the  cost  of  4500  feet  of  pine  boards  at  $22.50  per  M. 

2.  Find  the  cost  of  75  boxes  cigars,  each  containing  50,  at  $15.50  per  M. 

3.  What  will  1112  pounds  of  hay  cost  at  $1.40  per  cwt,? 

4-  Find  the  cost  of  a car  load  of  bran,  weighing  27350  lbs.,  at  65  cents  a cwt. 

5.  Find  the  cost  of  insuring  a property  valued  at  $7250  at  87J  cents  per 

$100. 

6.  Find  the  cost  of  40  boxes  envelopes,  each  containing  500,  at  $2.16§ 
per  M. 

7.  What  are  17610  feet  of  oak  lumber  worth  at  $27.25  per  M ? 

8.  Find  the  cost  of  the  following  merchandise: 

2175  lbs.  Middlings  at  $.65  a cwt. 


875  “ 

Cracked  Corn 

“ .50 

200  “ 

Oat  Meal 

“ 1.25 

3125  “ 

Bran 

“ .38 

9.  Find  the  cost  of  the  following  invoice  of  lumber : 

26281  ft.  Hemlock  at  $12.25  per  M. 

4278  “ Scantling  “ 12.50  “ 

7500  Shingles  “ 17.50  “ 

10.  What  is  the  cost  of  a car  load  of  hay,  18750  lbs.,  at  $13.50  per  ton  ? 

11.  Find  the  freight  on  hay  in  preceding  problem  at  8 cents  a C. 

12.  13560  pounds  of  wheat,  purchased  at  $1.37  a cental,  were  sold  at  $1  a 
bushel  of  60  lbs.  What  amount  was  gained  or  lost  by  the  transaction? 

13.  Find  the  cost  of  1812  pounds  of  hay  at  $14.50  a ton. 

11/..  What  is  the  value  of  65  pounds  of  wheat  at  $1.25  a cental  ? 

15.  1 sold  a bill  of  hemlock  lumber,  585622  feet,  and  gained  $1.30  a thous- 
and. What  was  my  entire  gain  ? 


QUANTITY,  PRICE  AND  COST 


95 


368.  To  find  cost  when  merchandise  is  sold  by  the  ton. 

Example. — What  is  the  value  of  5 tons  6 cvvt.  of  coal  at  $5.75  per  ton  ? 

5 tons  6 cwt. =5.30  tons. 


369.  Rule. — Multiply  the  hundredweights  by  .05  and  annex 
to  the  tons;  multiply  by  the  price  per  ton  and  point  off  required 
places. 

Note. — The  above  rule  applies  to  both  the  “short  ” or  standard  ton  of  2000 
pounds,  and  the  “ long  ” ton  of  2240  pounds,  the  latter  being  accompanied  by  the 
long  hundred  of  112  pounds,  which  equals  or  .05  of  a ton. 


/s-.7s 

S3 

/ 7 2.  s 
i r 7.r 

/3  O. 4 7 s 


PROBLEMS 

370.  1.  Find  the  cost  of  11  tons  18  cwt.  of  coal  at  $3  25. 

2.  Find  the  cost  of  9 tons  2 cwt.  of  coal  of  $5.50. 

3.  What  is  the  value  of  2 cars  of  coal,  one  containing  11  tons  3 cwt.,  the 
other  13  tons  15  cwt.,  at  $2.85  per  ton? 

Make  the  extensions  in  the  following  : 

1 car  Pea  Coal,  11  tons  6 cwt.  at  $3.25 

1 “ Egg  Coal,  12  tons  19  cwt.  “ 4.75 

1 “ Stove  Coal,  10  tons  13  cwt.  “ 4 50 

1 “ “ “ 13  tons  “ 4.50 

Total 

5.  What  is  the  value  of  18  tons  17  cwt.  of  iron  ore  at  $18.75  per  ton  ? 

6.  What  is  the  freight  on  43  tons  7 cwt.  of  iron  at  $1.35  a ton  ? 

7.  A man  bought  his  year’s  supply  of  coal,  12  tons  6 cwt.,  at  $5.75  ; what 
was  the  amount  of  the  bill  ? 

8.  Three  cars  of  iron  weighed  respectively  25  tons  7 cwt.,  23  tons  18  cwt., 
and  27  tons  2 cwt. ; what  was  the  value  of  the  iron  at  $14.80  a ton? 

9.  What  would  be  the  freight  on  the  iron  in  preceding  problem  at  35  cents 
a ton  ? 

10.  How  many  tons  of  2000  pounds  in  25  tons  9 cwt.  long  tons? 

371.  To  find  the  cost  of  commodities  sold  by  the  standard  or  “ short  ” 
ton  of  2000  pounds,  when  the  weight  is  given  in  pounds. 

Examples. — (1)  Find  the  cost  of  2840  pounds  of  hay  at  $18  a ton  ; (2)  of  3500 
pounds  of  bran  at  $25  a tou  ; (3)  of  1960  pounds  of  fertilizer  at  $32.50  a ton. 


2r 

s/o 

2 

P 4 

/S 

s/ 

0. 

^ ' fo) 


7 r. 


3S\ 

r 


0 O - 

VML- 


7S- 


s 


/f 

/ 0 

f 

ro 

2 

s/S 

f3/ 

ss 

fff  2 O . 

/ O.  , frj 

2.5 0 // 

2.5 0 // 


Since  1 cent  a pound  gives  $20  a ton,  and  since  most  ton  prices  can  be 
easily  figured  upon  that  basis,  use  the  following 

372.  Rule. — Point  off  two  decimal  places  in  the  number  of  pounds,  which  gives 
$20  a ton ; then  add  to  or  subtract  from  this  result  such  part  of  it  as  the  price  is  more 
or  less  than  $20. 


96 


QUANTITY,  PRICE  AND  COST 


ORAL  EXERCISE 


373.  Analyze  the  following  prices  upon  a basis  of  $20  a ton  and  apply  them 
to  such  quantities  as  may  he  given  by  the  teacher  : 


1.  $10  a ton. 

2.  $30  “ “ 

3.  $16  “ “ 
ff  $24  “ “ 

5.  $22  “ “ 

6.  $15  “ “ 


7.  $12.50  a ton. 

8.  $17.50  “ “ 

9.  $22.50  “ “ 

10.  $27.50  “ “ 

11.  $40  “ “ 

12.  $35  “ “ 


13.  $45  a ton. 
74.  $50  “ “ 

15.  $47.50“  “ 

16.  $37,50  “ “ 

17.  $21.50  “ “ 

18.  $65  “ “ 


374.  To  find  the  cost  of  commodities  sold  by  the  bushel  of  so  many 
pounds. 

Note. — It  is  evident  that  1 cent  a pound  for  any  commodity  sold  by  the  bushel,  is  as  many 
cents  a bushel  as  there  are  pounds  in  a bushel  of  the  commodity  in  question.  Thus,  1 cent  a pound 
equals  60  cents  a bushel  for  wheat,  clover  seed,  potatoes,  beans,  etc.  (these  commodities  each  weighing 
60  pounds  to  the  bushel) ; it  also  gives  32  cents  a bushel  for  oats,  45  cents  for  timothy  seed,  56  cents 
for  shelled  corn. 


Examples. — (1)  Find  the  cost  of  14680  lbs.  wheat  at  90c.  a bu.;  (2)  of  4368  lbs. 
oats  at  40c.  a bu. 


/ iff  6 

7 3 


2 0 


ro  = 
Ho_  = 


h 0 
3 Of 


v.  " ...  ...  A 


2 0 = fOff 


it  3 

/ O 

?2 

fSff 

(0  0 

= 32  f 

- rt 


o 


375.  Rule. — Point  off  two  decimal  places  in  the  quantity  and  increase,  or 
dimmish  this  value  (as  the  price  may  require ) by  the  use  of  aliquot  parts. 

376.  To  find  cost  without  changing  to  bushels. 


WRITTEN  EXERCISE 


1.  24370  lbs.  wheat 

at  $0.75  a bu.  11. 

29640  lbs.  oats 

at 

0 

-f 

0 

c/it 

2.  54940 

a tt 

U 

.90  “ 12. 

13664 

<<  U 

t< 

.45 

3.  94580 

tt  tt 

u 

.971  “ 13. 

9900 

a u 

tt 

.50 

4.  73690 

u u 

u 

1.05  “ Iff 

97980 

“ shelled  corn 

tt 

.70 

5.  66460 

it  tt 

t( 

1.03  “ 15. 

57428 

U it  tt 

it 

.60 

6.  9720 

“ beans 

u 

1.50  “ 16. 

7245 

“ timothy  seed 

tt 

1.05 

7.  17230 

tt  It 

u 

2.70  “ 17. 

6440 

“ barley 

tt 

.78 

8.  8850 

“ peas 

u 

1.95  “ 18. 

5280 

“ millet 

tt 

1.35 

9.  3190 

“ clover  seed 

u 

6.90  “ 19. 

12240 

“ rye 

it 

.84 

10.  2775 

u u u 

tt 

7.50  “ 20. 

99750 

“ corn  in  ear 

tt 

.42 

QUANTITY,  PRICE  AND  COST 


97 


REVIEW  PROBLEMS 

377.  1.  A farmer  hauled  to  market  25  loads  of  wheat  averaging  2796  lbs. 
to  the  load.  One-half  the  wheat  was  graded  as  “No.  1,”  for  which  he  received 
$1.05  a bushel;  4 as  “No.  2,”  for  which  he  was  paid  $1  a bushel;  and  the 
remainder  “ No.  3,”  worth  90  cents  a bushel.  How  much  did  he  get  for  his  wheat? 

Five  carloads  of  wheat,  weighing  respectively  52475,  54980,  46325,  49590 
and  51650  pounds,  were  shipped  from  Kansas  to  Philadelphia.  If  the  dealer 
who  shipped  the  wheat  paid  90  cents  a bushel  for  it,  274  cents  per  cwt.  for  freight, 
.and  $75  for  other  expenses,  and  then  sold  it  for  $1.20  a bushel,  what  was  his  profit? 

3.  Ten  loads  of  hay  weighing  respectively  2450,  2380,  2590,  2725,  2675, 
2295,  2885,  2565,  3020  and  2745  pounds,  were  sold  at  $22.50  a ton.  How  much 
did  he  receive  for  the  hay  ? 

A A stock  dealer  bought  125  head  of  hogs,  weighing  them  in  five  lots;  the 
first  lot  weighed  10500  lbs.,  the  second  lot,  9975  lbs.,  the  third,  11150  lbs.,  the 
fourth,  8995  lbs.  and  the  fifth,  9780  lbs.  What  was  the  average  weight  of  the 
hogs?  If  he  paid  6J  cents  a pound  for  them,  and  sold  them  at  cents  a pound 
after  paying  out  $105  for  freight  and  other  expenses,  what  was  his  profit? 

5.  A Pittsburg  grain  merchant  bought  245760  lbs.  of  shelled  corn  at  42c.  a 
bu.,  385600  lbs.  of  corn  in  the  ear  at  40c.  a bu.,  296440  lbs.  of  oats  at  36c.  a bu., 
and  582380  lbs.  of  wheat  at  93c.  a bu.  What  was  the  entire  cost? 

6.  If  the  merchant  mentioned  in  the  preceding  problem  had  the  corn  in  the 
ear  shelled,  losing  thereby  4 of  its  weight  in  the  cobs,  but  disposing  of  the  cobs 
at  $5  a ton,  then  selling  all  the  corn  at  49c.  a bu.,  the  oats  at  45c.  a bu.,  and  the 
wheat  at  $1.03  a bu.,  what  was  his  net  gain  on  the  whole,  his  expenses  amounting 
to  $593.45  ? 

7.  How  much  better  would  the  merchant  in  the  preceding  problem  have 
done  by  having  the  wheat  converted  into  flour,  at  an  expense  of  334c.  per  barrel 
for  the  milling  and  20c.  apiece  for  barrels,  and  then  selling  the  flour  at  $5.25  a 
bbl.  (196  lbs.),  the  bran  and  middlings  at  $22  a ton,  estimating  a bushel  of  wheat 
to  make  42  lbs.  of  flour  and  allowing  any  fraction  of  a barrel  of  flour  for  waste? 

8.  A New  Jersey  farmer  marketed  50  loads  of  potatoes  of  175  baskets  each. 
If  the  average  net  weight  of  the  potatoes  was  374  lbs.  to  the  basket  and  he 
received  75c.  a bushel  for  them,  what  did  he  realize  from  the  crop? 


ANALYSIS 


378.  Many  arithmetical  problems  may  be  solved  readily  by  analysis,  pro- 
ceeding from  the  known  to  the  unknown,  according  to  the  conditions  of  the 
problem,  using  the  unit  as  the  basis  of  comparison. 

MENTAL,  PROBLEMS 

Example. — What  will  be  the  cost  of  5 oranges,  if  3 oranges  cost  12  cents? 

Solution. — 1 orange  will  cost  4 cents  and  5 oranges  will  cost  20  cents. 

Note. — It  is  deemed  unnecessary  that  the  student  go  through  a long  formula  of  words  ; let  him 
use  sufficient  words  to  show  that  the  mental  trend  is  connected  and  logical,  and  require  no  more. 

379.  1.  What  will  6 lemons  cost  at  the  rate  of  8 lemons  for  16  cents? 

2.  What  is  the  value  of  7 pair  of  shoes,  if  4 pair  are  worth  $12? 

3.  If  7 men  can  mow  14  acres  of  grass  in  a day,  how  many  acres  can 
12  men  mow  in  the  same  time? 

1/..  What  are  8 oranges  worth,  if  9 oranges  cost  18  cents? 

5.  If  9 pounds  of  beef  cost  54  cents,  what  will  7 pounds  cost  ? 

6.  At  the  rate  of  6 barrels  of  flour  for  $30,  what  will  11  barrels  cost? 

7.  If  a man  travels  64  miles  in  8 days,  how  far  can  he  travel  in  14  days  ? 

8.  If  a drover  feeds  hay  at  the  rate  of  22  tons  in  11  weeks,  how  many  tons 
will  he  feed  in  7 weeks? 

9.  How  many  apples  can  be  bought  for  42  cents  at  the  rate  of  14  for 
7 cents  ? 

10.  A wheelman  rode  at  the  rate  of  84  miles  in  7 hours;  how  far  would  he 
ride  in  9 hours? 

11.  If  9 men  can  mow  18  acres  of  grass  in  a day,  how  much  can  11  men 
mow  in  the  same  time? 

1 2.  If  8 men  can  dig  24  rods  of  ditch  in  a day,  how  much  can  9 men  dig  in 
the  same  time? 

13.  If  7 boys  can  do  a piece  of  work  in  15  days,  how  long  will  it  take 
21  boys  to  do  it? 

11/..  How  many  men  will  be  required  to  build  ashed  in  3 days,  if  5 men  can 
do  it  in  12  days  ? 

15.  How  many  men  can  mow  as  much  grass  in  5 days  as  4 men  can  mow 
in  40  days  ? 

16.  If  it  requires  12  men  6 days  to  build  a wall,  how  many  men  will  be 
required  to  build  it  in  12  days  ? 

17.  If  8 yards  of  cloth  cost  $24,  what  will  f-  of  20  yards  cost  ? 

18.  If  a farmer  gave  9 bushels  of  wheat  for  2 barrels  of  flour,  what  was  the 
wheat  worth  a bushel  if  8 barrels  of  flour  cost  $72  ? 

19.  A farmer  exchanged  apples  worth  $2  a barrel  for  4 }Tards  of  cloth  which 
cost  at  the  rate  of  $21  for  7 yards ; how  many  barrels  of  apples  did  he  give  ? 

20.  If  f of  a yard  of  muslin  cost  9 cents,  what  will  2 yards  cost? 

Solution. — 1 of  a yard  will  cost  3 cents,  1 yard  will  cost  12  cents  and  2 yards  will  cost  24  cents. 


98 


ANALYSIS 


99 


21.  If  of  a box  of  tea  cost  $10,  what  will  1 box  cost? 

22.  What  will  3 boxes  of  soap  cost,  if  f of  a box  cost  $6  ? 

23.  If  | of  a yard  of  cloth  cost  $8,  what  will  3 yards  cost? 

24-  If  f of  a barrel  of  flour  cost  $4,  what  is  the  the  value  of  12  barrels? 

25.  If  | of  a yard  of  cloth  cost  $7,  what  will  8 yards  cost? 

26.  If  A can  walk  24  miles  in  -f-  of  a day,  how  far  can  he  walk  in  6 days  ? 

27.  If  f of  a box  of  raisins  cost  $5,  what  will  7 boxes  cost? 

28.  How  much  will  8 barrels  of  apples  cost  if  of  a barrel  cost  70  cents  ? 

29.  How  much  will  7 tons  of  hay  cost  if  $20  will  buy  f of  a ton  ? 

30.  A watch  cost  $42  and  f of  its  cost  was  twice  the  cost  of  a chain ; what 

was  the  cost  of  the  chain? 

31.  A horse  cost  $100  and  4 of  its  cost  was  the  cost  of  a carriage ; what  did 
both  cost  ? 

32.  If  11  yards  of  cloth  cost  $5J,  what  will  5 yards  cost? 

Solution. — §5£  are  $V-,  1 yard  will  cost  and  5 yards  will  cost  $4  or  $24 

33.  If  5 oranges  are  worth  7J  cents,  what  is  one  dozen  worth  ? 

34-.  What  cost  18  bananas  at  the  rate  of  1\  cents  for  5 bananas? 

35.  If  11  chickens  cost  $4f , what  will  15  chickens  cost  ? 

36.  If  J of  16  yards  of  cloth  cost  $34,  wThat  will  4 of  18  yards  cost? 

37.  If  -g-  of  42  yards  of  muslin  cost  49  cents,  what  will  § of  36  yards  cost  ? 

38.  If  3 apples  cost  f of  a cent,  what  will  10  apples  cost? 

39.  If  8 pair  of  shoes  cost  $-2A,  what  will  10  pair  cost  ? 

40.  If  3 shaddocks  cost  f of  1 dollar,  what  will  8 shaddocks  cost? 

41.  How  much  are  12  lamps  ivorth,  if  7 lamps  are  worth  $^g4-? 

42.  What  will  f-  of  15  yards  of  cloth  cost,  if  4 of  24  yards  cost  4 of  $36  ? 

43.  What  is  the  value  of  6 mirrors,  if  9 mirrors  are  worth  $^-? 

44 ■ A boy  is  12  years  old,  which  is  J his  father’s  age ; what  is  the  father’s 
age? 

Solution. — If  12  years  is  4 the  father’s  age,  the  father  is  4 times  12  years,  or  48  years  old. 

45.  A sheep  cost  $8,  which  wras  f the  cost  of  a cow ; what  would  3 cows  cost? 

46.  James  has  20  pennies,  which  is  § of  John’s  number ; how  many  have  both  ? 

47.  Martha’s  hat  cost  $7,  which  was  4 of  the  cost  of  her  dress ; what  was  the 
cost  of  both  ? 

48.  A clerk  spent  $20,  which  is  4 of  the  sum  he  has  left ; how  much  had  he 
at  first  ? 

49.  A man  earned  $30,  which  is  f of  what  he  has  in  a savings  bank  ; how 
much  has  he  in  bank  ? 

50.  A watch  was  bought  for  $25,  which  is  4 of  5 times  what  the  chain  cost  \ 
what  was  the  cost  of  both  ? 

51.  The  head  of  a fish  is  5 inches  long,  which  is  4 of  twice  the  length  of  the 
body  ; what  is  the  length  of  the  body  ? 


100 


ANALYSIS 


52.  Wheeler  paid  $5  for  a saddle,  which  is  4 of  J of  the  cost  of  his  wheel ; 
what  did  his  wheel  cost? 

53.  The  head  of  a fish  is  4 inches  long  and  the  tail  5 inches  long,  which 
together  is  f of  J of  the  length  of  the  body  ; how  long  is  the  fish? 

54 . Thomas,  who  is  15  years  old,  is  •§-  of  Harry’s  age ; how  old  is  Harry  ? 

Solution. — £ of  Harry’s  age  is  \ of  15  years  or  3 years  ; Harry  is  6 times  3 years  old,  or  18 
years  old. 

55.  A woman  found  $12,  which  was  f of  what  money  she  then  had;  how 
much  had  she  at  first? 

56.  A farmer  sold  a cow  for  $32,  which  was  f of  the  cost  of  the  cow  ; what 
did  the  cow  cost? 

57.  A drover  sold  a horse  for  $120,  thereby  gaining  4 of  its  cost ; wThat  was 
its  cost  ? 

58.  Leary  sold  a book  for  80  cents,  which  was  4 of  -f  of  its  cost  when  new ; 
what  was  its  cost? 

59.  Harkness  sold  a horse  for  $140,  which  was  § of  4 of  its  value;  what  was 
the  horse  worth  ? 

60.  18  feet  of  a pole  is  in  the  water,  which  is  f of  -f  of  the  length  in  the  air  ; 
what  is  the  length  of  the  pole? 

61.  A pole  stands  40  feet  in  the  air,  which  is  J-of  f of  the  length  of  the  pole; 
what  is  the  length  of  the  pole? 

62.  A horse  cost  $150,  and  § of  the  cost  of  the  horse  is  twice  the  cost  of  the 
harness;  what  was  the  cost  of  both? 

63.  A vest  cost  $5,  which  is  f of  of  the  cost  of  a coat  and  J of  f the  cost  of 
trousers;  what  did  the  suit  cost? 

64-  A cow  cost  $35,  which  was  4 of  twice  the  cost  of  a horse  ; what  did  both 
cost  ? 

65.  A boy’s  pony  cost  $44,  which  was  -f  of  f-  of  the  cost  of  his  wagon  ; what 
was  the  cost  of  his  wagon  ? 

66.  If  5 boys  can  earn  $6f  in  a week,  how  much  can  6 boys  earn  in  the 
same  time  ? 

67.  How  far  will  a man  walk  in  3 hours  if  in  7 hours  he  walks  194  miles? 

68.  If  a man  earns  $54  in  3 days,  how  much  can  he  earn  in  4 days? 

69.  If  4J  tons  of  coal  cost  $27,  how  much  will  7 tons  cost  at  the  same  rate? 

70.  How  much  will  a man  earn  in  a week  at  the  rate  of  $24  a day  ? 

71.  If  3 loads  of  hay  cost  $154,  what  will  6 loads  cost? 

72.  If  a vessel  can  sail  23  miles  in  4J-  hours,  how  far  can  it  sail  in  9 hours  ? 

73.  If  a man  rides  14  miles  in  2J  hours,  how  far  will  he  ride  in  4f  hours? 

74 ■ If  4 cows  eat  2 tons  of  hay  in  8 weeks,  how  long  will  the  same  amount 

last  5 cows? 

75.  If  2 men  can  do  a piece  of  work  in  16  days,  how  long  will  it  take  8 men 
to  do  the  same  work  ? 


ANALYSIS 


101 


76.  A watch  cost  $50,  which  is  f of  3 times  what  the  chain  cost.  Required 
the  cost  of  both. 

77.  A man  having  § of  a barrel  of  flour  bought  f of  a barrel ; how  much 
had  he  then  ? 

78.  A man’s  money  increased  by  its f equals $90  ; how  much  money  had  he? 

79.  A man  owned  f of  a boat  and  sold  l of  the  boat ; what  part  of  the  boat 
did  he  still  own  ? 

80.  The  difference  between  J-  of  my  money  and  of  my  money  is  $10  ; 
what  is  the  amount  of  my  money  ? 

81.  A boy  having  36  pennies,  lost  f of  them  and  then  found  -§  as  many  as 
he  had  at  first ; how  many  had  he  then  ? 

82.  Margaret  has  $12  and  her  brother  has  f-  as  much  ; how  much  have  both  ? 

83.  f of  is  jig-  of  the  cost  of  a chain  ; how  much  did  the  chain  cost? 

8 If..  A hat  cost  f of  $10,  which  is  J-  of  the  cost  of  a coat.  Required  the  cost 
of  both. 

85.  If  5 pints  of  milk  cost  13  cents,  how  many  pints  of  milk  can  you  buy 
for  26  cents  ? 

86.  If  12  sheep  cost  $25,  what  will  15  sheep  cost  at  the  same  rate? 

87.  A chain  cost  $12  and  J of  its  cost  is  f of  the  cost  of  a watch  ; what  is  the 
cost  of  both  ? 

88.  If  a yard  of  cloth  costs  § of  $1,  how  many  yards  can  be  bought  for  $10  ? 

89.  How  many  yards  of  cloth  can  be  bought  for  $6,  if  f of  $1  buys  2 yards? 

90.  If  6 men  can  do  a piece  of  work  in  6-f  days,  how  long  will  it  take  5 men 
to  do  it  ? 

91.  A man  being  asked  his  age  said,  that  if  to  his  age  its  4 and  its  § were 
added,  he  would  be  52  years  old  ; what  was  his  age? 

92.  A man  having  lost  f of  his  money  had  $24  remaining;  how  much  did 
he  lose  ? 

93.  f of  the  length  of  a pole  is  in  the  air,  J in  the  water,  and  the  remainder, 
18  feet,  in  the  earth ; what  is  the  length  of  the  pole  ? 

91f.  What  number  is  that  which  being  increased  by  its  J,  and  that  sum 
diminished  by  its  §,  the  remainder  is  30? 

95.  What  is  a man’s  money  if  twice  his  money  diminished  by  $12  equals  $88  ?' 

96.  A man’s  money  increased  by  $80  equals  three  times  his  money;  how 
much  money  has  he? 

97.  If  a tree’s  height  be  increased  by  § its  height  and  20  feet,  the  sum  will 
equal  twice  its  height;  what  is  its  height? 

98.  A man  and  a boy  earned  $24  in  a week ; what  did  each  earn  if  the  man 
earned  twice  as  much  as  the  boy? 

99.  A watch  and  chain  cost  $70  ; what  was  the  cost  of  each  if  of  the  cost  of 
the  watch  equals  the  cost  of  the  chain  ? 

100.  A pole  40  feet  long  was  broken  in  two  unequal  parts  so  that  f of  the 
longer  equals  the  shorter  ; what  is  the  length  of  each  part  ? 


102 


ANALYSIS 


WRITTEN  PROBLEMS 

380.  Example. — If  5 men  reap  ten  acres  of  wheat  in  a day,  how  many  acres 
will  7 men  reap  in  the  same  time? 

Solution. — If  five  men  reap  ten  acres  of  wheat  in  a day,  one  man  will  reap  one-fifth  as  much  or 
two  acres,  and  seven  men  will  reap  fourteen  acres. 

Example — If  10  men  dig  a ditch  in  4 days  of  10  hours  each,  in  how  many 
days  will  12  men  dig  a similar  ditch,  working  8 hours  a day  ? 

Solution. — If  10  men  dig  it  in  4 days  of  10  hours  each,  it  would  require  one  man  40  days  of  10 
hours,  or  400  hours  ; 12  men  would  do  it  in  -jj  of  400  hours,  or  33J  hours  ; and  working  eight  hours  a 
day  would  require  days. 

Example. — A can  do  a piece  of  work  in  10  days,  B in  12  days;  in  what  time 
can  both  working  together  do  it? 

Solution. — A can  do  r'ff  of  it  in  one  day  ; B can  do  in  one  day  ; both  working  together  can 
do  XV  and  x\,  or  H in  one  day.  They  can  do  f$,  the  whole  piece  of  work,  in  5t5j  days. 

381.  1 ■ If  a post  5 feet  high  casts  a shadow  12  feet  long,  how  high  is  a 
telegraph  pole  which  at  the  same  time  casts  a shadow  65  feet  long? 

A pipe  will  fill  a cistern  in  10  hours,  another  pipe  in  6 hours  ; if  both  be 
opened  at  the  same  time,  how  long  will  be  required  to  fill  the  cistern  ? 

3.  A man  can  do  a piece  of  wor  k in  6 days,  and  a boy  can  do  it  in  10  days. 
In  what  time  can  they  do  it  by  working  together? 

J.  An  inlet  pipe  will  fill  a tank  in  6 hours,  and  an  outlet  pipe  will  empty 
it,  when  full,  in  10  hours.  If  both  be  left  open  when  the  tank  is  empty,  how  long 
will  be  required  to  fill  the  tank  ? 

5.  Jones  can  dig  a ditch  in  12  days,  Smith  in  15  days,  and  Brown  in  18 
days.  How  long  will  be  required  for  them  to  dig  a ditch  twice  as  deep,  if  they 
all  work  together? 

6.  B can  do  ^ as  much  work  in  a day  as  A.  How  long  would  it  require 
both  working  together  to  do  that  which  B alone  can  do  in  20  days? 

7.  Green  can  unload  a car  of  corn  in  6 hours;  Black  can  unload  it  in  5 
hours.  How  many  hours  would  both  require,  working  together,  to  unload  3 
similar  cars  ? 

8.  A can  do  a piece  of  work  in  16  days.  B in  20  days.  If  both  together 
begin  the  work,  and  A leaves  when  it  is  f done,  how  long  will  B require  to 
complete  it  ? 

9.  If  I buy  goods  at  4 yards  for  $5  and  sell  at  the  rate  of  5 yards  for  §7, 
how  many  yards  must  be  sold  to  gain  $70? 

10.  If  10  men  can  do  a piece  of  work  in  24  days,  and  4 men  retire  when  the}' 
have  worked  10  days,  in  what  time  will  the  remaining  men  finish  the  work  ? 

11.  If  5 men  or  9 boys  can  do  a piece  of  work  in  12  days,  in  how  many  days 
should  9 men  and  5 boys  do  the  same  work? 


ANALYSIS 


103 


12.  10  men  agreed  to  do  a piece  of  work,  but  2 of  them  did  not  report,  on 
account  of  which  the  work  took  3 days’  more  time.  In  what  time  could  the  10 
men  have  done  the  work? 

13.  A cau  carry  a ton  of  coal  to  the  fourth  story  of  a building  in  \ of  a day, 
B in  f of  a day,  and  C in  •§■  of  a day.  How  long  will  it  require  A,  B and  C, 
together,  to  carry  1 ton  ? 

Ilf..  A,  B and  C can  do  a piece  of  work  in  8,  10  and  15  days  respectively. 
After  the  3 have  worked  2 days,  how  long  will  it  require  B and  C together  to 
finish  it? 

15.  A does  a piece  of  work  in  J day,  B in  § of  a day,  C in  2 days,  and  D in 
If  days.  How  long  will  it  require  to  perform  the  work  if  they  all  work  together  ? 

16.  If  a loaf  of  bread  weighs  8 ounces  when  flour  is  $3.75  a barrel,  what 
should  it  weigh  wdien  flour  is  $4.50  a barrel? 

17.  If  I gain  $1250  by  selling  $20000  worth  of  goods,  how  much  should  I 
sell  to  gain  $6250? 

i/  18.  If  9|  pounds  of  tea  cost  $6.75,  what  would  174  pounds  be  worth  ? 
i/  19.  If  $7.60  will  buy  144  pounds  of  soap,  how  many  pounds  can  be  bought 
for  $19.75  ? 

20.  If  114  acres  of  land  will  pasture  33  cows,  how  many  acres  will  be 
required  for  327  cows  ? 

21.  If  6 men  can  do  a piece  of  work  in  17  days,  how  many  men  would  be 
required  to  do  the  work  in  11  days? 

22.  If  a cycler  can  ride  225  miles  in  2 4 days  of  8 hours  each,  how  many 
days  of  ten  hours  each  would  be  required  to  ride  625  miles  ? 

23.  If  6 men  can  mow  a field  in  10  hours,  how  many  men  must  be  added  to 
mow  it  in  6 hours  ? 

21f.  If  a man  can  do  4 of  a piece  of  work  in  6 days  of  10  hours  each,  in  what 
time  can  he  do  the  remainder,  working  8 hours  a day? 

25.  A,  B and  C own  a vessel,  A owning  § and  each  of  the  others  half  the 
remainder ; B sells  f of  his  share  for  $3000 ; what  is  the  value  of  A’s  and  C’s 
share  at  the  same  rate  ? 

26.  If  8 horses  or  7 cows  eat  6f  tons  of  hay  in  23  days,  how  long  will  it  take 
7 horses  and  8 cows  to  eat  the  same  quantity  ? 

27.  If  15  men  can  do  a piece  of  work  in  10  days,  how  long  will  be  required 
to  do  it  if  10  men  quit  when  the  work  is  half  done  ? 

28.  One  pump  can  fill  a cistern  in  4 hours  and  another  in  5 hours,  and  an 
outlet  pipe  can  empty  it  in  1 hour ; if  the  tank  be  empty  and  the  pumps  both 
working,  in  what  time  can  it  be  filled? 

29.  If  the  tank  in  the  preceding  problem  be  full  with  both  pumps  working 
and  the  outlet  pipe  open,  in  what  time  will  the  tank  be  emptied  ? 


104 


ANALYSIS 


30.  A can  mow  an  acre  in  5J  hours,  and  A and  B together  can  mow  twice  as 
much  in  4^  hours  ; in  what  time  can  B working  alone  mow  2J  acres? 

31.  Two  men  start  in  business  with  the  same  amount  of  capital ; the  first 
gains  \ of  his  capital  and  the  second  gains  \ of  his  capital,  when  the  first  has 
$100  more  than  the  second.  What  was  the  capital  of  each  ? 

32.  C can  reap  a field  of  grain  in  6 days  and  D in  8 days.  In  how  many 
days  can  they  do  it  reaping  together? 

33.  A cistern  has  two  pipes,  lyythe  first  of  which  it  may  be  filled  in  12  hours, 
and  by  the  second  in  8 hours;  how  long  will  both  be  in  filling  it? 

3 If..  D and  E working  together  can  make  a china  closet  in  8 days ; D work- 
ing alone  could  do  it  in  12  days;  how  long  would  it  take  E to  make  it? 

35.  A owns  J of  a tract  of  land  and  B owns  y1^.  A’s  share  is  worth  $250 
more  than  B’s.  Find  the  value  of  the  tract. 

36.  A and  B agree  to  do  a piece  of  work,  A to  receive  $2  a day  and  B $3  a 
day.  A works  twice  as  many  days  as  B,  and  they  together  receive  $70.  How 
many  days  does  each  labor  ? 

37.  I sold  a book  and  lost  £ of  its  cost.  Had  it  cost  $1  less,  I would  have- 
gained  | of  its  cost.  What  was  its  cost? 

38.  A man  sold  a book,  making  £ of  the  cost.  Had  the  book  cost  $1  less  and 
sold  for  the  same  profit,  he  would  have  gained  three  times  the  cost.  Find  the 
cost. 


39.  A and  B invest  equal  sums  of  money ; A gains  £ of  his  investment  and 
B loses  $20;  B’s  money  is  then  equal  to  £ of  A’s.  How  much  does  each  invest?" 

IfO.  A can  do  a piece  of  work  in  5 days ; B can  do  it  in  6 days.  In  how 
many  days  can  they  do  it  working  together? 

Ifl.  G can  reap  a field  of  grain  in  6§  days  and  D in  8f  days.  In  how  many 
days  can  they  do  it  reaping  together? 

If.2.  A tank  has  two  pipes,  by  the  first  of  which  it  may7  be  filled  in  12 
hours,  and  by  the  second  in  15  hours;  how  long  will  both  be  in  filling  it? 

If3.  E and  F working  together  can  make  a china  closet  in  5 days;  E work- 
ing alone  could  do  it  in  9 days;  how  long  would  it  take  F to  make  it? 

Iflf.  A boy  hired  to  a mechanic  for  twenty  weeks  on  condition  that  he  should 
receive  $20  and  a coat.  At  the  end  of  twelve  weeks  the  boy  ceased  work,  when 
it  was  found  that  he  was  entitled  to  $9  and  the  coat.  What  was  the  value  of 
the  coat  ? 


MEASURES 

Measures  of  Value 

382. 

U.  S.  Money 

10  Mills  = 1 Cent 

E $ d ct.  m 

1 = 10  = 100  = 1000  = 10000 

10  Cents  = 1 Dime 

1 = 10  = 100  = 1000 

10  Dimes  = 1 Dollar 

1 = 10  = 100 

10  Dollars  = 1 Eagle 

1 = 10 

383. 

English  Money 

4 Farthings  = 1 Penny 

£ s.  d.  f. 

1 = 20  = 240  = 960 

12  Pence  = 1 Shilling 

1 = 12  = 48 

20  Shillings  = 1 Pound 

1 = 4 

£ 1 = $4.8665*  (Gold) 

384. 

French  Money 

100  Centimes  = 1 Franc 

Fr.  Cent. 

1 = 100 

1 Franc  = $0,193*  (Gold). 

385. 

German  Money 

100  Pfennigs  = 1 Mark 

M.  Pf. 

1 = 100 

1 Mark  = $0,238*  (Gold). 


Other  Foreign  Moneys 


Notes. — The  Latin  Union  countries — France,  Belgium,  Switzerland,  Greece  and  Italy  have  the 
same  monetary  unit,  called  the  “ franc”  in  the  first  three,  the  “drachma”  in  Greece,  and  the“lira  ” 
in  Italy.  The  standard  gold  and  silver  coins  circulate  freely  in  the  different  countries. 

Finland,  Spain  and  Venezuela  have  monetary  units  of  the  same  value  as  the  franc  of  France 
($0,193),  called  respectively  “mark,”  “peseta’’  and  “bolivar.” 

Denmark,  Norway  and  Sweden  have  a like  monetary  unit  called  the  “crown”;  value  $0,268. 

Mexico,  Guatemala,  Honduras,  Nicaragua,  Salvador,  Chile,  Uruguay,  Argentine  Republic  and 
the  Philippine  Islands  have  each  a monetary  unit  known  as  the  “ peso,”  varying  in  value  according  to 
the  pure  silver  or  gold  contents  of  the  coins  of  those  countries,  the  lowest  being  tliat  of  Chile,  worth 
$0,365,  and  the  highest,  that  of  Uruguay,  worth  $1,034.  (The  silver  coins  fluctuate  in  value,  while 
the  gold  coins  do  not. ) 

Besides  the  United  States,  the  following  six  countries  or  colonies  have  the  “dollar  ” as  a mone- 
tary unit,  the  value  of  which  is  appended  in  each  case  : British  Honduras  ($1.00),  Canada  ($1.00), 

Columbia  ($1.00),  Hong-Kong  ($0,463),  Liberia  ($1.00),  Newfoundland  ($1,014).  These  coins  are  all  of 
gold,  except  that  of  the  British  colony  of  Hong-Kong,  which  is  of  silver  valued  as  above  on  April 
1,  1908. 

* These  monetary  unit  values  are  determined  by  the  Director  of  the  United  States  mint,  and 
proclaimed  in  a circular  issued  every  three  months  by  the  Secretary  of  the  Treasury.  They  repre- 
sent the  intrinsic  value  of  the  precious  metal  in  the  coins  and  are  used  in  estimating  the  value  of 
imported  merchandise  upon  which  duties  are  collected  at  the  various  custom  houses.  They  have  noth- 
ing to  do  with  the  commercial  rates  at  which  bills  of  exchange  are  bought  and  sold,  the  latter  fluctua- 
ting with  the  supply  and  demand  in  the  foreign  exchange  market.  The  commercial  rates  may,  there- 
fore, be  higher  or  lower  than  these  intrinsic,  or  par  value  rates. 


105 


106 


MEASURES  OF  EXTENSION  AND  SURFACE 


Measures  of  Extension  and  Surface 


386.  Long 

12  Inches  = 1 Foot 
3 Feet  = 1 Yard 
54  Yards  = 1 Rod 
320  Rods  = 1 Mile 


M EASURE 

Mi.  rd.  yd.  ft.  in. 

1 = 320  = 1760  = 5280  = 63360 
1=  5i=  16J=  198 

1 = 3 = 36 

1 = 12 


39.37  Inches  = 1 Meter 

A knot  is  equal  to  6087  feet. 

The  hand,  used  in  measuring  the  height  of  horses,  equals  4 inches. 

A pace  is  equal  to  three  feet,  and  5J  paces  approximate  a rod. 

A rod  is  sometimes  called  a perch  or  pole. 

A statute  mile  in  the  United  States  and  England  is  5280  feet,  and  is  distinguished  from  the  geo- 
graphic mile  which  is  6087  feet. 

The  yard  is  the  standard  unit  of  extension  of  the  United  States  and  also  of  Great  Britain. 

387.  Surveyors’  Long  Measure 


7.92  Inches  1 Link 

25  Links  1 Rod 

100  Links  (4  rods)  1 Chain 

80  Chains  1 Mile 


The  unit  of  surveyors’  long  measure  is  the  Gunter’ s chain,  which  is  4 rods,  or  66  feet  in  length. 
Since  the  chain  has  100  links,  links  may  be  written  as  hundredths  of  a chain.  Thus  18  chains,  22 
links  = 18.22  chains. 


388.  Square 

144  Square  Inches  = 1 Square  Foot 
9 Square  Feet  =1  Square  Yard 
30|  Square  Yards  = 1 Square  Rod 
160  Square  Rods  = 1 Acre 
640  Acres  = 1 Square  Mile 


Measure 

A.  sq.  rd.  sq.  yd.  sq.  ft.  sq.  in. 

1 = 160  = 4840  = 43560  = 6212640 

1 = 30i  = 2721  _ 39182 

1 = 9 = 1296 

1 = 144 


1.196  Square  Yards  = 1 Square  Meter 

With  the  exception  of  the  acre,  the  units  of  square  measure  are  derived  from  the  corresponding 
units  of  long  measure. 

389.  Surveyors’  Square  Measure 

625  Square  Links  = 1 Square  Rod 

16  Square  Rods  = 1 Square  Chain 

10  Square  Chains,  or  160  Square  Rods  = 1 Acre 
640  Acres  = 1 Square  Mile 

36  Square  Miles  = 1 Township 

The  unit  of  land  measure  is  the  acre. 

In  surveying  public  lands,  a square  mile,  640  acres,  is  called  a “ section.” 


MEASURES  OF  VOLUME  OR  CAPACITY 


107 


Measures  of  Volume  or  Capacity 

390.  Solid  Measure 

1728  Cubic  Inches  = 1 Cubic  Foot 
27  Cubic  Feet  = 1 Cubic  Yard 

128  Cubic  Feet  = 1 Cord 
40  Cubic  Feet  = 1 Ton  (for  freight) 
24f  Cubic  Feet  = 1 Perch  of  Stone. 
(16*  X 1*  X l = 24f.) 


cu.  yd.  cu.  ft.  cu.  in. 

1 = 27  = 46656 
1 - 1728 


1.308  Cubic  Yards  = 1 Cubic  Meter 

1 Cubic  Meter  or  Stere  is  the  metric  unit  of  Wood  Measure. 

The  unit  of  Lumber  Measure  is  a Board  Foot,  which  is  T\  of  a Cubio  Foot. 


391.  Liquid  Measure 

4 Gills  = 1 Pint 
2 Pints  = 1 Quart 
4 Quarts  = 1 Gallon 


gal.  qt.  pt.  gi. 

1 = 4 = 8 = 32 

1 = 2=  8 

1 = 4 


The  Wine  Gallon  contains  231  cubic  inches. 

For  estimating,  a barrel  contains  31*  gallons,  and  a hogshead  2 barrels. 
Commercially,  the  exact  contents  of  barrels  are  found  by  gauging. 

1.05668  Liquid  Quarts  = 1 Liter  (used  for  milk  and  wine) 


392.  Dry  Measure 


2 Pints  = 1 Quart 
8 Quarts  = 1 Peck 
4 Pecks  = 1 Bushel 


bu.  pk.  qt.  pt. 

1 = 4 = 32  = 64 

1 = 8 =16 

1 = 2 


A bushel,  stricken  measure  (the  Winchester  bushel)  contains,  approximately,  2150.4  cubic  inches, 
and  is  used  in  measuring  grain,  such  as  wheat,  corn  (shelled),  beans  ; also,  small  fruits,  chestnuts,  etc. 

A heaped  bushel  contains,  approximately,  2747.7  cubic  inches,  and  is  used  in  measuring  large 
fruits  (apples,  pears,  etc.),  vegetables,  corn  in  the  ear,  coal,  coke,  lime,  etc. 

The  British  Imperial  bushel  contains,  approximately,  2218.2  cubic  inches. 

The  dry  gallon  or  half  peck,  contains  268.8  cubic  inches,  or  * of  2150.4. 

.9081  Dry  Quarts  = 1 Liter  100  Liters  = 1 Hectoliter 

2.8375  Bushels  = 1 Hectoliter 


108 


MEASURES  OF  WEIGHT 


Measures  of  Weight 

393.  Troy  Weight 

24  Grains  = 1 Pennyweight 

20  Pennyweights  = 1 Ounce 
12  Ounces  = 1 Pound 


lb.  oz.  pwt.  gr. 

1 = 12  = 240  = 5760 

1 = 20  = 480 

1 = 24 


The  Troy  pound,  5760  grains,  is  the  unit  of  weight.  Troy  weight  is  used  in  weighing  gold,  silver 
and  precious  stones. 

The  carat,  for  weighing  diamonds,  is  3.2  Troy  grains. 

The  carat,  indicating  fineness  of  gold,  means  A part.  Gold  20  carats  fine  has  20  parts  gold  and  4 
parts  alloy. 

394.  Apothecaries’  Weight 

it.  § 3 9 gr. 

20  Grains  = 1 Scruple  1 = 12  = 96  = 288  = 5760 

3 Scruples  — 1 Dram  1 = 8 = 24  = 480 

8 Drams  - 1 Ounce  1 = 3 = 60 

12  Ounces  = 1 Pound  1 = 20 


This  weight  is  used  in  weighing  small  quantities  of  drugs  and  medicines.  For  large  quantities 
avoirdupois  weight  is  used. 

395.  • Avoirdupois  Weight 

T.  cwt.  lb.  oz. 

16  Ounces  = 1 Pound  1 = 20  = 2000  = 32000 

100  Pounds  = 1 Hundredweight  1 = 100  = 1600 

20  Hundredweights  = 1 Ton  1 = 16 


The  gross  or  long  ton  used  at  U.  S.  Custom  Houses,  and  by  wholesale  dealers  in  coal  and  iron,  is 
2240  pounds,  the  hundredweight  112  pounds,  the  quarter  28  pounds. 

1 Pound  Avoirdupois  = 7000  Troy  Grains 


15.432354  Troy  Grains 

2.2046  Pounds  Av.  (approximately 

2204.6  Pounds  “ 

1000  Grams 
1000  Kilos 


■ 1 Gram. 

= 1 Kilogram. 

= 1 Ton  or  Tonneau. 
= 1 Kilogram 
= 1 Ton  or  Tonneau. 


396.  W eights  of  Grains 


Article 

Measure 

Pounds 

Exceptions 

Article 

Measure 

Pounds 

Exceptions 

Barley 

bushel 

48 

Pa.  47 

Hemp 

bushel 

44 

Beans 

bushel 

60 

Millet 

bushel 

50 

Buckwheat 
Clover  Seed 

bushel 

bushel 

50 

60 

Pa.  48 
N.  J.  64 

Oats 

bushel 

32 

( N.  J.  30 
\ Md.  26 

Corn  in  ear 

bushel 

70 

Peas 

bushel 

60 

Corn,  shelled 

bushel 

56 

Rye 

bushel 

56 

Flaxseed 

bushel 

56 

N.  J.  55 

Timothy  Seed 

bushel 

45 

Wheat 

bushel 

60 

The  above  table  and  the  one  following  contain  the  weights  of  the  common  grains,  fruits,  and 
vegetables,  as  used  by  a majority  of  the  States.  Exceptions  are  given  in  Pa.,  Md.  and  N.  J. 


MEASURES 


109 


397.  Weights  of  Fruits,  Vegetables,  Etc. 


Article 

Measure 

Pounds 

Exceptions 

Article 

Measure 

Pounds 

Apples 

bushel 

50 

N.  J.  48 

Beef 

barrel 

200 

Peaches,  dried 

bushel 

33 

Pork 

barrel 

200 

Potatoes 

bushel 

60 

Pa.  56 

Flour 

barrel 

196 

Sweet  Potatoes 

bushel 

54 

Fish 

quintal 

100 

Peanuts 

bushel 

22 

Grain 

cental 

100 

Corn  Meal 

bushel 

50 

Nails 

keg 

100 

Salt,  coarse 

bushel 

70 

Pa.  85 

Lime,  unslacked 

bushel 

80 

Onions 

bushel 

57 

Pa.  50 

398. 


Time  Measure 


60  Seconds 
60  Minutes 
24  Hours 
7 Days 
365  Days 

12  Calendar  Months 


= 1 Minute 
= 1 Hour 
= 1 Day 
= 1 Week 

= 1 Year 


Add  one  day  for  Leap  Years.  All  years  exactly  divisible  by  4,  excepting  Centennial  Years,  are 
Leap  Years.  Centennial  Years  exactly  divisible  by  400  are  Leap  Years. 


399.  Circular  Measure 

60  Seconds  = 1 Minute 
60  Minutes  = 1 Degree 
360  Degrees  = 1 Circle 

This  table  is  used  for  measuring  angles  and  arcs,  and  for  latitude  and  longitude. 
The  signs  are  degree  (°),  minute  ('),  second  ("). 


Longitude  and  Time 

400.  As  the  circle  of  the  earth,  360°,  passes  under  the  sun  in  24  hours,  in 
one  hour  ^4  of  the  circle,  or  15°,  would  pass;  hence,  15°  of  longitude  make  a 
difference  of  1 hour  in  time;  hence,  to  find  difference  in  longitude,  multiply 
difference  in  time  by  15;  and  to  find  difference  in  time,  divide  difference  in 
longitude  by  15. 


401.  Miscellaneous 


20  Units  = 1 Score 
12  Units  = 1 Dozen 
12  Dozen  = 1 Gross 
12  Gross  = 1 Great  Gross 


24  Sheets  = 1 Quire 
20  Quires  = 1 Ream 


6 Feet  =1  Fathom 

1|-  Miles  = 1 Knot 

3 Knots  = 1 League 

60  Knots  or 

Geographical  Miles 
69-jt  Statute  Miles 


1 Degree 


DENOMINATE  NUMBERS 


402.  Denominate  numbers  are  concrete  numbers  expressing  divisions  of 
money,  weight,  measure,  etc.  They  may  be  reduced,  added,  subtracted,  multi- 
plied and  divided. 

403.  To  reduce  to  lower  denominations. 

Example. — Reduce  £13  9s.  lOd  to  pence. 

20  12 
£ s.  d. 

13  9 10 

£13  9S.  I0d.  = 3238d. 

269 

12 

3238 

404.  Rule. — Multiply  the  number  of  higher  denomination  by  the  number  of  the 
next  lower  denomination  in  a unit  of  the  higher,  adding  the  number  of  the  lower 
denomination  to  the  product.  Proceed  until  the  required  denomination  is  reached. 

WRITTEN  EXERCISE 

405.  1 Reduce  18  lb.  11  oz.  3 pwt.  16  gr.  to  grains. 

2.  Reduce  12  lb.  5s  63  19  14  gr.  to  grains. 

3.  Reduce  77  bu.  1 pk.  1 pt.  to  pints. 

4-  Reduce  12  mi.  30  rd.  5 yd.  2 ft.  10  in.  to  inches. 

5.  Reduce  365  da.  5 hr.  48  min.  49  sec.  to  seconds. 

6.  Reduce  77°  26'  45"  to  seconds. 

7.  Change  9 tons  12  cwt.  60  lb.  12  oz.  to  ounces. 

8.  Reduce  145  mi.  124  rd.  4 yd.  to  feet. 

9.  Change  215  gal.  3 qt.  1 pt.  to  pints. 

10.  How  many  pence  in  £472  10s.? 

406.  To  reduce  or  change  to  higher  denomination. 

Example. — Reduce  7534  pt.  to  bushels. 

2)7534 

8)3767 

4)  470+  i qt.  7534  pt.  =117  bu.  2 pk.  7 qt. 

117  + 2 pk. 

407.  Rule. — Divide  the  lower  denomination  by  the  number  in  a unit  of  the  next 
higher  denomination,  the  remainder,  if  any,  being  put  down  as  a part  of  the  result. 
Proceed  until  the  required  higher  denomination  is  obtained. 

no 


DENOMINATE  NUMBERS 


111 


WRITTEN  EXERCISE 

408.  1 ■ Change  1213  pt.  to  higher  denominations. 

2.  Change  478260  gr.,  apothecaries’  weight,  to  higher  denominations. 

3.  Change  95346  gr.,  Troy  weight,  to  higher  denominations. 

If.  Change  288692  oz.,  avoirdupois  weight  to  higher  denominations. 

5.  Change  4841  far.  to  higher  denominations. 

6.  Change  1456  gi.  to  gallons. 

7.  Change  436614  in.  to  miles. 

8.  Change  18500  sq.  ft.  to  higher  denominations. 

9.  Change  108000  seconds  of  time  to  higher  denominations. 

10.  Change  160000  seconds  of  circular  measure  to  higher  denominations. 

409.  To  reduce  denominate  fractions  and  decimals. 

Example  1. — Reduce  \ mi.  to  feet. 

80 

m ii 

lx  As  = 880  ft. 

f 1 2 

2 

Example  2. — Reduce  | ton  to  lower  denominations. 

5 50 

Zx2°  = - = 174  ix^  = 50 

8 2 2 1 

2 Result,  17  cwt.  50  lb. 


Example  3. — Reduce  .255  bu.  to  lower  denominations. 


.255 

4 

1.020 

8 

.16 

2 

.32 


Result,  1 pk.,  0.32  pt. 


Example  4. — Reduce  f ft.  to  fraction  of  a rod. 


I 

jL 


xl='  rd. 
11  22 


112 


DENOMINATE  NUMBERS 


Example  5. — Change  .3759  pt.  to  decimal  of  a gallon. 

2)  .3759 
4)  .18795 

.0469875  gal. 


. — Reduce  8 oz. 

7 pwt. 

20 

24 

oz. 

pwt. 

gr. 

8 

7 

12 

20 

167 

24 

680 

1 lb. =5760  gr. 

4020  _ 201  _ 67  lb 
5760  288  96 


334^ 

4020  gr. 

g7 

Example  7. — The  same  expressed  decimally:  ~g  = -6979| 


WRITTEN  EXERCISE 

410.  1 . Change  of  a mile  to  the  decimal  of  a rod. 

2.  Reduce  -1173410  of  a ton  to  the  decimal  of  a pound. 

3.  Change  .007  of  an  acre  to  the  decimal  of  a square  rod. 
If..  Change  ^ of  a gross  to  the  decimal  of  a dozen. 

5.  Reduce  of  a gallon  to  lower  denominations. 

6.  Reduce  of  a square  mile  to  lower  denominations. 

7.  Change  yjVo  °f  a pound  to  the  decimal  of  a scruple. 

8.  What  decimal  of  a ton  is  .00625  of  an  ounce  ? 

9.  Change  .025  bu.  to  lower  denominations. 

10.  Change  f-  of  a foot  to  the  decimal  of  a rod. 

11.  Change  § of  a pound  to  the  decimal  of  a ton. 

12.  What  decimal  of  a cubic  yard  is  430  cu.  in.? 

13.  What  decimal  of  a pound  sterling  is  5 d.? 

Ilf..  Change  } of  a pint  to  the  decimal  of  a bushel. 

15.  What  decimal  of  an  entire  circumference  is  40"? 

16.  What  decimal  of  a degree  is  52J'? 

17.  What  part  of  a square  rod  is  4J  sq.  yd.? 

18.  What  decimal  of  an  acre  is  6^  sq.  ft.? 

19.  What  part  of  a mile  is  an  inch? 

20.  Change  yt  of  a mile  to  lower  denominations. 

21.  What  is  the  value  of  ^ of  a bu.  in  pints? 

22.  Reduce  T9T  of  a square  mile  to  lower  denominations. 


DENOMINATE  NUMBERS 


113 


23.  If  § of  a pound  of  gold  is  \yorth  $174.25.  what  is  the  value  of  § of  a 
pennyweight  ? 

Sit..  A man  had  lOf  tons  of  coal  after  buying  2.5  tons ; how  many  pounds 
had  he  at  first? 

25.  From  f of  a pound  sterling  take  .05  of  a shilling. 

26.  What  part  of  1.25  miles  is  150  rd.  2 yd.  6 in.  ? 

27.  Reduce  65  65  29  to  the  fraction  of  a pound. 

28.  Reduce  4 oz.  10  pwt.  20  gr.  to  the  decimal  of  a pound. 

29.  Reduce  14  hr.  50  min.  30  sec.  to  the  decimal  of  a day. 

30.  Change  10s.  6d.  to  the  fraction  of  a pound  sterling. 

31.  What  part  of  3 pk.  6 qt.  is  4 qt.  1 pt.  ? 

32.  What  part  of  6 score  is  14  dozen  ? 

33.  What  part  of  a cubic  yard  is  2 cu.  ft.  280  cu.  in.  ? 

34-.  What  part  of  8^  bu.  is  2 pk.  64  qt.  ? 

35.  If  a man  earns  80  cents  in  2 hr.  40  min.,  what  will  he  earn  in  6 days  if 
he  works  8 hours  a day  ? 


411.  To  add  denominate  numbers. 

Example. — Find  the  sum  of  26  bu.  3 pk.  4 
bu.  5 qt. 

4 8 

bu.  pk.  qt. 

26  3 4 

18  1 3 

17  0 5 

62  1 4 


qt. ; 18  bu.  1 pk.  3 qt.;  and  17 


Result,  62  bu.  1 pk.  4 qts. 


412.  Rule. — Find  the  sum  of  the  lowest  denomination,  reduce  this  sum  to  next 
higher,  setting  down  remainder,  if  any  ; carry  the  number  of  next  higher  denomina- 
tion, find  sum,  and  reduce  in  like  manner  ; so  continue  until  addition  is  completed. 


413.  To  subtract  denominate  numbers. 

Example. — From  31  gal.  2 qt.  1 pt.  take  15  gal.  3 qt. 

gal.  qt.  pt. 

31  2 1 

2 - g q Result,  15  gals.  3 qts.  1 pt. 

15  3 1 


414.  Rule. — Find  the  difference  of  lowest  denomination  ; if  the  subtrahend  num- 
ber is  greater  than  the  minuend  number,  borrow  one  of  the  next  higher  denomination, 
adding  to  the  minuend  number  as  many  of  the  lower  denomination  as  make  one  of  the 
higher,  then  find  difference.  After  borrowing,  diminish  the  minuend  number  of  the 
next  higher  denomination  by  one,  then  find  difference.  So  continue  until  the  subtrac- 
tion is  completed. 


114 


DENOMINATE  NUMBERS 


415.  To  multiply  denominate  numbers. 

Example. — Multiply  3 yd.  2 ft.  5 in.  by  120. 

3 12 

5 X 120  = 600  in.  = 50  ft, 

yd-  ln-  2 X 120  = 240  + 50  = 290 ft,= 

3 2 5 96  yd.  2 ft. 

120  3 X 120  = 360  + 96  = 456  yd. 

Result  456  yd.  2 ft. 

456  2 0 

416.  Rule. — Multiply  as  in  whole  numbers  and  reduce  to  next  higher,  setting 
down  remainder,  if  any,  and  carrying  the  number  of  higher  to  product  of  next  higher. 
Proceed  in  like  manner  until  full  product  is  obtained. 

417.  To  divide  denominate  numbers. 

Example — Divide  176°  42'  12"  by  12. 

522  372 

12)176°  42'^  12" 

14°  43'  31" 

418.  Rule  .—Divide  the  number  of  highest  denomination  and  set  down  the  quo- 
tient ; reduce  the  remainder,  if  any,  to  next  lower  denomination,  add  to  result  the  number 
in  the  dividend  of  same  denomination,  set  down  quotient  and  proceed  in  like  manner 
until  division  is  completed.  If  the  lowest  denomination  divided  has  a remainder,  it 
may  be  expressed  fractionally. 

419.  The  methods  of  adding,  subtracting,  multiplying  and  dividing  here 
exemplified  are  not  much  used  in  commercial  operations.  Custom  usually  des- 
ignates some  denomination  as  the  one  to  be  used,  and  quantities  are  named  in  it 
and  fractional  parts  of  it. 


WRITTEN  PROBLEMS 

420.  1.  If  a book  cost  6d,  how  many  may  be  bought  for  3£  5s.  6d.  ? 

2.  Change  192000  oz.  to  tons. 

3.  A haystack  contains  23000  lb.;  what  is  its  value  at  $15  a ton  ? 
f.  How  many  grains  Troy  in  12  lb.  4 oz.  12  pwt.? 

5.  How  many  2 drachm  powders  of  magnesia  can  be  put  up  from  13  lb. 
5 oz.  ? 

6.  What  is  the  cost  of  three  bushels  of  tomatoes  at  12  cents  a quart? 

7.  Reduce  15  gal.  1 qt.  1 pt.  to  gills. 

8.  How  many  acres  of  land  in  a tract  containing  2S774S0  sq.  ft.  ? 

9.  How  many  half-acre  lots  in  a piece  of  ground  containing  S200  sq.  rd. 
30  sq.  yd.?  State  the  surplus,  if  any,  in  square  feet. 

10.  What  is  the  value  of  a diamond  weighing  of  a carat  at  $75  a carat? 

11.  What  is  the  value  of  the  pure  gold  in  an  ornament  weighing  4 pwt.,  18 
carats  fine,  at  $20  an  ounce  for  pure  gold? 

12.  Reduce  80  pounds  Avoirdupois  to  denominations  of  Troy  weight, 

13.  Change  224  oz.  Troy  to  denominations  of  Avoirdupois  weight. 

If.  What  is  the  value  of  20000  pounds  of  oats  at  35  cents  a bushel  ? 


DENOMINATE  NUMBERS 


115 


15.  How  many  yards  of  cloth  at  7s.  3 d a yard  can  be  bought  for  £8  10s.? 

16.  What  is  the  value  of  £11  5s.  4 d at  $4.8665  a pound  sterling? 


Notk. — A business  rule  for  changing  shil- 
lings and  pence  is  to  multiply  the  shillings  by  5, 
and  point  off  2 places  ; the  pence  by  4|  , point  oil  3 
places. 

Suggestion. 

1 s.  = £ 2ir  or  .05. 

1 d.  = £370  or  -004^. 


17.  What  is  the  value  in  francs  of  £25  7s.  3d  ? 

18.  What  is  the  value  in  United  States  money  of  220  yards  broadcloth  at 
8s.  5 d a yard  ? 

19.  What  is  the  value  in  English  money  of  $1200? 

20.  Bought  250  meters  silk  at  5.10  francs  a meter  and  sold  it  at  $1.25  a yard. 
How  much  was  gained  ? 

21.  A ship  sails  7512  kilometers  in  60  days.  How  many  knots  does  it 
average  per  hour  ? 

22.  What  would  be  the  cost  of  excavating  a cellar  12  meters  long,  6 meters 
wide,  2.5  meters  deep,  at  36  cents  a cubic  yard  ? 

23.  How  many  cords  of  wood  in  a pile  30  feet  long,  6 feet  high,  the  sticks 
being  4 feet  long  ? 

24-  A cask  of  wine  contains  160  liters  worth  a franc  a liter.  How  much 
will  be  gained  or  lost  by  selling  it  at  a dollar  a gallon  ? 

25.  The  five-cent  nickel  piece  weighs  5 grams;  how  many  could  be  coined 
from  a kilo  of  the  metal  ? 

26.  Change  f-J  of  a mile  to  lower  denominations. 

27.  Change  £.2685  to  lower  denominations. 

28.  Reduce  £11  4s.  2 Id  to  pounds  sterling. 

29.  What  decimal  of  an  acre  is  112  sq.  rd.  ? 

30.  How  many  slates  covering  2f  sq.  ft.  each  will  cover  a square  of  100 
sq.  ft.  ? 

31.  How  many  reams  of  paper  will  be  needed  to  print  an  edition  of  20000 
copies  of  a book  containing  216  pages,  each  sheet  folding  into  8 leaves? 

32.  How  many  degrees  in  f of  a circle?  How  many  minutes?  How  many 
seconds? 

33.  What  does  1 of  the  circumference  of  a circle  measure  in  degrees,  minutes 
and  seconds  ? 


^ //  


dr 


4- 'A 


-2qr 

o / 6>% 


o / 


/ / . A 4 4 

u.r  6,  6 


o 


r / 3 3 
4 y ^ 4 
4 7 s f 6 
<70/2? 

4 '-S  O 6 4 

3-2.  V V 3 

z f 23  3 3 


Res.  $54.83. 


116 


DENOMINATE  NUMBERS 


34-  What  part  of  a circumference  is  16°  18'  25"? 

35.  Find  the  time  by  compound  subtraction  from  the  date  of  the  Declaration 
of  Independence  until  to-day.  Reduce  to  minutes,  counting  30  days  to  the 
month. 

36.  If  the  equator  is  25000  miles  long,  what  is  the  length  of  10  degrees? 

37.  What  is  the  length  of  a quadrant,  if  a degree  measures  4 inches? 

38.  Give  the  length  of  a degree  in  a sextant,  when  the  circumference  meas- 
ures 12  ft.  6 in. 

39.  How  many  scores  in  160  dozen? 

4.0.  How  many  quires  in  500  reams  ? 

41.  Buy  pens  at  $1.25  a gross  and  sell  them  at  a cent  apiece.  What  will  be 
the  gain  in  a sale  of  a great  gross? 

4%.  How  long  has  a sum  of  money  been  on  interest,  that  was  placed  July  4, 
1876?  (Time  by  compound  subtraction.) 

43.  How  many  cubic  feet  in  a cistern  that  will  hold  135  barrels? 

44-  How  many  cubic  feet  in  a bin  that  will  hold  200  bushels  of  wheat? 

45.  What  is  the  difference  in  time  between  a place  in  longitude  69°  50'  west 
and  a place  120°  25'  west? 

46.  Change  -^111 0 of  a degree  to  the  decimal  of  a minute. 

47.  What  decimal  of  a square  rod  is  | of  a square  foot? 

48.  If  f of  a pound  of  gold  is  worth  $172.50,  what  is  the  value  of  .375  pwt.  ? 

49.  What  would  a man  gain  by  selling  by  the  short  ton  at  the  same  price 
that  he  had  paid  in  purchasing  by  the  long  ton  ? 

50.  Find  what  fractional  part  of  a bushel  3 pk.  6 qt.  is,  and  change  the  result 
to  a decimal. 

51.  Bought  172.5  meters  silk  at  5.18  francs  per  meter  and  sold  it  at  $1.93  a 
yard.  How  much  was  gained? 

52.  Buy  wheat  at  12.35  francs  a hectoliter  and  sell  it  at  a dollar  a bushel. 
How  much  is  gained  on  a hectoliter? 

53.  What  would  be  gained  by  buying  wine  at  1 franc  a liter  and  selling  it 
for  20  cents  a quart? 

54-  Bought  100  kilos  French  candy  at  5.18  francs  a kilo  and  sold  it  for  half 
a dollar  a pound.  What  wTas  my  gain  or  loss  on  a pound  ? 

55.  What  part  of  a tonneau  would  he  gained  by  buying  coal  by  the  long 
ton  and  selling  by  the  tonneau  at  the  same  price? 

56.  What  part  of  a ton  would  be  lost  by  buying  coal  by  the  tonneau  and 
selling  it  by  the  long  ton  at  the  same  price? 

57.  How  many  square  rods  in  a piece  of  land  containing  7 square  chains? 

58.  If  the  circumference  of  the  earth  is  24899  miles,  how  many  kilometers 
is  it? 

59.  If  the  circumference  of  earth  is  25000  miles,  what  is  the  length  of  a 
degree? 


PRACTICAL  MEASUREMENTS 

421.  A line  has  one  dimension  only — length. 

422.  A surface  has  two  dimensions — length  and  width. 

423.  A square  unit  is  a square  in  which  ihe  units  of  length  and  width  are 
alike. 

424.  A solid  has  three  dimensions — length,  width  and  thickness. 

425.  A cubic  unit  is  a cube  in  which  the  units  of  length,  width  and  thick- 
ness are  alike. 


SURFACE  MEASUREMENTS 


Note. — Under  this  head  come  such  practical  measure- 
ments as  carpeting,  plastering,  papering,  painting,  roofing, 
flooring,  paving,  land  measurements,  etc. 

426.  To  find  the  area  of  rectangular  sur- 
faces. 


RECTANGLE 


Example. — How  many  square  feet  in  a tight  board  fence  6 feet  high  and  40 
rods  long? 


40  rods  = 660  ft. 

660  ft.  X 6 = 3960  sq.  ft. 

Note. — When  the  dimensions  are  expressed  in  different 
denominations,  they  should  be  reduced  to  units  of  the  same 
denomination  before  multiplying;  as,  inches  to  feet  , or  feet  to 
inches. 

427.  Rule. — Multiply  the  length  by  the  width  ; 
the  product  will  be  the  area. 


WRITTEN  PROBLEMS 

428.  1.  What  is  the  value  of  a lot  32  feet  front,  120  feet  deep,  at  25  cents  a 
square  foot  ? 

2.  How  many  square  yards  of  linoleum  will  be  required  to  cover  a floor 
22J  feet  long,  14J  feet  wide? 

3.  What  will  be  the  cost  of  painting  a wainscoting  40  feet  in  length,  3 feet 
6 inches  high,  at  25  cents  a square  yard  ? 


117 


118 


PRACTICAL  MEASUREMENTS 


f A railroad  60  feet  wide  passes  half  a mile  through  a tract  of  land.  If 
the  damage  is  assessed  at  $150  an  acre,  what  must  the  company  pay? 

5.  What  will  it  cost,  at  $2.25  a square  yard,  to  pave  with  asphalt  a street 
50  feet  wide  for  a length  of  1600  feet  ? What  must  a property  owner  pay  who 
has  a frontage  of  40  feet  ? 

6.  How  many  square  feet  of  tin  will  be  required  to  cover  a flat  roof  40  feet 
long  and  20  feet  wide  ? What  will  be  the  cost  at  $5.25  per  square  of  100 
square  feet? 

7.  At  $67.50  per  thousand  square  feet,  what  will  it  cost  to  floor  and  ceil  a 
room  60  feet  long  and  25  feet  wide  ? 

8.  A ceiling  18  feet  10  inches  long,  14  feet  7 inches  wide  may  be  of  hard 
wood,  costing  12Jc.  a square  foot,  or  of  corrugated  iron  at  $1.35  a square  yard. 
Which  is  the  cheaper  and  how  much  ? 

9.  The  floor  of  a church  80  feet  long,  45  feet  wide,  is  covered  with  matting. 
Find  the  cost  of  cleaning  it  at  3 cents  a square  yard. 

10.  A bill-board  78  feet  6 inches  long  and  8 feet  high  was  made  at  a cost  of 
6Jc.  a sq.  ft.  What  was  the  cost  ? 


ROOFING  AND  PAVING 

429.  To  find  the  number  of  bricks,  shingles,  slates,  etc.,  required  to 
cover  a given  surface. 

Examples. — (1)  Find  the  number  of  bricks  4x8  inches  required  to  lay  a 
walk  5 feet  wide  and  40  feet  long ; (2)  Find  the  number  of  pieces  each  2 inches 
square  required  to  inlay  a vestibule  floor,  6 feet  8 inches  by  5 feet  6 inches. 


/o 


/t 


S X ^Q-X  - <?oo 

4^  X 

7 (7  O 


Suggestion  For 
Example  2 
6 ft.  8 in.  = 80  in.  long 
5 ft.  6 in.  = 66  in.  wide 


33 

t^X  lH~  = / 3 2, 0 
•5> 


430.  R ule. — Divide  the  given  surface  by  the  surface  of  a unit  to  find  the 
required  number. 


Note. — The  work  may  he  greatly  shortened  by  cancelation,  as  shown  in  examples  above. 


PRACTICAL  MEASUREMENTS 


119 


WRITTEN  PROBLEMS 

431.  1-  How  many  shingles,  averaging  4 inches  in  width,  laid  five  inches  to 
the  weather,  will  be  required  to  cover  two  sides  of  a roof  40  feet  long,  each  side  18 
feet  wide  ? What  will  the  shingles  cost  at  $12  per  M ? 

Note. — The  ordinary  shingles  are  16  in.  by  4 in.  and  are  put  up  in  bundles  of  250  each.  Noth- 
ing less  than  an  entire  bundle  should  be  considered  if  purchased  by  the  bundle. 

2.  At  $12.50  per  M,  what  will  the  shingles  cost  for  a roof  60  ft.  long,  18ft. 
from  eaves  to  ridge  on  one  side,  and  24  ft.  on  the  other,  the  shingles  being  4 
inches  wide,  laid  4J  inches  to  the  weather  ? At  $3  per  bundle? 

3.  How  many  slates  each  10  inches  wide,  laid  5J  inches  to  the  weather  will 
be  required  for  a roof  50  ft.  long  and  20  ft.  from  eaves  to  ridge  ? 

4-  How  many  bricks,  8X4,  laid  flat,  will  be  required  to  pave  a 6-foot  walk 
800  feet  long  ? What  will  the  bricks  cost  at  $14  a thousand  ? What  will  the 
laying  cost  at  60  cents  a square  yard  ? 

5.  What  will  it  cost  to  pave  a section  of  street  40  feet  wide,  600  feet  long, 
wiih  vitrified  brick  laid  on  edge,  9X24  in.,  at  $11  per  thousand  and  72  cents  a 
square  yard  for  laying  ? 

6.  How  many  wooden  blocks  each  2J  in.  thick,  6 in.  wide  and  10  in.  long, 
placed  on  end,  will  it  take  to  pave  a street  50  ft.  wide,  for  a distance  of  2J  miles. 
What  will  the  paving  cost  at  $4.75  a sq.  yd.  for  material  and  labor? 


PLASTERING,  PAPERING  AND  PAINTING 


432.  To  find  entire  surfaces  of  rooms,  boxes,  etc. 


Formula  : (Lengthq-Width)X2=Perimeter. 

P.  XHeight=Surface  of  Walls. 

L.XW.=Surface  of  top  or  bottom. 

Note. — In  problems  on  plastering,  etc.,  the  surface  of  the 
walls+the  surface  of  the  ceiling,  divided  by  9 gives  the  number 
of  square  yards.  If  any  allowance  is  to  be  made  for  the  space, 
occupied  by  doors  and  windows,  deduct  such  allowance  before 
reducing  to  square  yards. 


yj  r el  i v 

— 1 FT.— J—  1 FT.  -J—  1 ft.— 

3 

F 

E 

E 

T 

1 SQUARE  YARD 

Example. — How  many  square  yards  of  plastering  in  the  walls  and  ceiling 
of  a room  24  feet  long,  16  feet  wide  and  12  feet  6 inches  high  above  the  base- 
board, no  allowance  for  doors  and  windows? 


120 


PRACTICAL  MEASUREMENTS 


433.  Rule — Multiply  the  perimeter  ( distance  around ) by  the  height,  and  to  this 
resu  lt  add  twice  the  product  of  the  length  by  the  width. 


/3  r # 


<?j  / 3 r# 


WRITTEN  PROBLEMS 

434.  1.  How  many  square  feet  of  surface  has  a box  that  is  5 ft.  6 in.  long, 
3 ft.  6 in.  wide  and  4 ft.  high  ? 

2.  How  many  square  feet  of  surface  has  a room  that  is  18  feet  long,  14  feet 
wide,  9J  feet  high  ? What  would  be  the  cost  of  plastering  the  walls  and  ceiling 
at  35  cents  a square  yard,  no  allowance  for  doors  and  windows  ? 

3.  At  42  cents  a square  yard,  what  will  it  cost  to  plaster  the  walls  and  ceil- 
ing of  a hallway  80  ft.  long,  9 ft.  wide  and  15  ft.  high,  allowing  for  half  the  space 
occupied  by  12  doors,  each  5 ft.  4 in.  wide  and  9 ft.  high? 

4-  Find  the  cost  of  lathing,  plastering  and  flooring  a hall  30  ft.  by  50  ft. 
and  20  ft.  high  at  85  cents  a sq.  yd.  for  the  lathing,  35  cents  a sq.  yd.  for  plaster- 
ing, and  $4.50  per  100  sq.  ft.  for  the  flooring ; allowance  to  be  made  in  the  lathing 
for  the  full  space  occupied  by  12  windows,  each  5 ft.  wide  and  8 ft.  6 in.  high,  and 
2 doors  6 ft.  6 in.  by  12  ft.,  and  § of  the  door  and  window  space  to  be  allowed 
in  the  plastering. 

5.  How  many  rolls  of  paper  8 yards  long,  18  inches  wide,  would  be  required 
for  the  sides,  ends  and  ceiling  of  a room  15  ft.  6 in.  long,  14  ft.  3 in.  wide  and  9 
ft.  high  ? 

Note. — Single  rolls  of  wall  paper  are  8 yards  long,  double  rolls,  16  yards  long.  Paper  hangers 
estimate  the  number  of  rolls  required  by  measuring  the  unbroken  wall  space  ; it  is  found  that  the 
irregular  spaces  above  doors  and  windows  can  be  covered  from  the  remnants  of  the  rolls. 

6.  What  will  be  the  cost  of  papering  the  sides  and  ends  of  a room  18  feet 
long,  15  feet  wide,  10  feet  high,  deducting  10  feet  from  the  perimeter  for  doors 
and  windows,  if  the  paper  costs  50  cents  a double  roll  ? 

7.  Estimate  the  cost  of  papering  the  walls  and  ceiling  of  5 rooms,  each  24 
ft.  long,  16  ft.  wide  and  12  ft.  high,  the  baseboard  being  12  in.  high  and  the 
border  18  in.  wide.  Six  inches  are  allowed  to  each  strip  for  matching,  and  4 
single  rolls  are  to  be  deducted  for  doors  and  windows  in  each  room.  The  paper 
costs  90  cents  a double  roll,  and  the  border  12J  cents  a yard.  The  work  can  be 
done  by  2 men  in  lj  days,  at  $2.75  each  per  day. 


PRACTICAL  MEASUREMENTS 


121 


CARPETING 

435.  Carpet  is  sold  by  the  yard,  a yard  of  carpet  being  3 feet  long,  regard- 
less of  its  width.  Linoleum  and  oil  cloth  are  generally  sold  by  the  square  yard. 

In  estimating  the  number  of  yards  of  carpet  for  a room  it  is  always  neces- 
sary to  find  the  number  of  strips  required,  and  to  do  this  we  must  know  whether 
they  are  to  run  lengthwise  or  crosswise.  They  are  generally  laid  lengthwise, 
but  if  the  room  is  nearly  square  they  may  run  either  way,  and  sometimes  there 
is  a considerable  saving  in  laying  them  crosswise. 

436.  To  find  the  number  of  yards  of  carpet  required  for  a room. 

437.  Rule. — If  the  strips  are  to  be  laid  lengthwise,  divide  the  width  of  the  room 
in  inches  by  the  width  of  the  carpet  in  inches  ; the  result  will  be  the  number  of  strips, 
counting  any  fraction  as  a whole  strip. 

If  the  strips  are  to  be  laid  crosswise,  divide  the  length  of  the  room  in  inches 
by  the  width  of  the  carpet  in  inches,  and  count  any  fraction  left  over  as  a whole  strip. 

Multiply  the  length  of  the  strips  in  feet,  plus  any  allowance  for  waste  in  matching, 
by  the  number  of  strips,  and  divide  by  3. 


Example. — Which  will  be  the  cheaper,  and  how  much,  to  lay  the  strips 
lengthwise  or  crosswise  on  a floor  18  ft.  by  20  ft.,  carpet  27  in.  wide,  costing  $1.50 
a yard,  if  there  is  no  waste  in  matching  either  way  ? 


Carpet  laid  crosswise : 


/ 

20Fee 

ir8% 

T = 24 

Strips 

IV/OE 

O/nchl 

27/ncnt 

fl 

r— 

% 

\ 

t 

** 

Sf 

hi 

s 

y> 

< 

0, 

<5> 

18X12=  8 strips,  each  20  ft.  6f  yds.  long. 

27  6f  yds.X8=53j  yds. 

Or,  8 strips  each  20  ft.=160  ft. 

160  ft. -=-3=53 J yds. 

Note. — The  heavy  lines  in  the  diagrams 
indicate  the  direction  of  the  strips,  and  the 
dotted  lines  the  number  of  yards  in  each  strip. 


>0X12=^: 

w 

9 


: 8|  strips,  or  9 strips,  each 
18  ft.,  or  6yds.  long=54 
yds. 


54  yds. — 53}  yds.=f  yds.  less  length- 
wise. 

f-  yds.  at  $1.50=$1.00  the  result. 


122 


PRACTICAL  MEASUREMENTS 


WRITTEN  EXERCISE 


438. 

Find  the  number  of  yards  and  th 

e cost  of  each  of  the  following: 

Length 

Width  Width 

Strips  Waste 

Price 

of 

of  of 

to  in 

of 

Room. 

Room.  Carpet. 

Run.  Matching. 

Carpet. 

1. 

25  ft. 

19  ft.  27  in. 

Lengthwise  None 

$1.50 

2. 

40  ft. 

31  ft.  27  in. 

Lengthwise  6 in. 

3.25 

3. 

28  ft. 

27  ft.  27  in. 

Lengthwise  1 ft. 

2.50 

If. 

20  ft. 

18  ft.  27  in. 

Crosswise  None 

1.25 

5. 

35  ft. 

22  ft.  36  in. 

Lengthwise  9 in. 

.95 

6. 

19  ft.  6 in. 

15  ft.  9 in.  27  in. 

Cheaper  way  None 

1.10 

7. 

72  ft.  9 in. 

40  ft.  6 in.  36  in. 

Lengthwise  1 ft.  3 in. 

1.12i 

8. 

50  ft. 

40  ft.  27  in. 

Cheaper  wTay  1 ft. 

1.50 

9. 

13  ft.  6 in. 

12  ft.  27  in. 

Crosswise  10  in. 

1.75 

10. 

16  ft. 

14  ft.  31J  in. 

Cheaper  way  3 in. 

1.33J 

11. 

30  ft.  8 in. 

21  ft.  11  in.  36  in. 

Lengthwise  4 in. 

1.25 

12. 

22  ft.  6 in. 

20  ft.  10  in.  27  in. 

Cheaper  way  None 

1.16f 

13. 

24  ft. 

18  ft.  31|  ju. 

Length  wflse  1 ft. 

1.35 

Ilf. 

27  ft. 

24  ft.  6 in.  27  in. 

Cheaper  way  None 

1.50 

15. 

11  ft.  3 in. 

10  ft.  9 in.  36  in. 

Crosswise  6 in. 

2.25 

WRITTEN  PROBLEMS 

439.  1.  How  many  yards  of  carpet  £-  yd.  wide  (27  in.)  will  be  required  for  a 
room  15  ft.  wide  and  22  ft.  long,  if  the  strips  are  laid  lengthwise  and  there  is  a 
waste  of  1 foot  on  each  strip  in  matching? 

2.  What  will  it  cost  at  87J  cents  a yard  to  carpet  a room  17  ft.  by  22  ft. 
6 in.,  the  carpet  being  1 yd.  wide,  and  laid  the  more  economical  way,  no  allow- 
ance for  waste  in  matching? 

3.  Find  the  cost  at  $1.12J  a yard  to  carpet  a room  17  ft.  6 in.  wide  and 
30  ft.  9 in.  long  with  carpet  27  in.  wide,  the  strips  to  be  laid  lengthwise  and  9 in. 
to  be  allowed  on  each  strip  for  waste  in  matching. 

If.  At  97 J cents  a yard  for  the  carpet  and  5 cents  a square  yard  for  lining, 
what  would  it  cost  to  cover  a floor  30  ft.  wide  and  50  ft.  long  with  carpet  314 
in.  wide,  no  allowance  for  waste  in  matching? 

5.  At  $1.50  a yard,  what  will  it  cost  to  make  a rug,  the  body  of  which  is  to 
be  3 yards  wide  and  6 yards  long,  made  of  27-inch  Axminster,  and  the  border 
surrounding  it  to  be  24  inches  wide,  with  no  waste  except  that  caused  by  the 
diagonal  folds  at  the  corners  of  the  border? 

6.  A lady  desiring  to  cover  the  floor  of  her  parlor,  20  feet  long  and  17  feet 
wide,  has  the  choice  of  three  kinds  of  carpet,  viz : 27  inches  wide,  with  an 
allowance  of  6 inches  for  matching,  at  624c.  a yard ; ^ of  a yard  wide,  with  S 
inches  allowance  for  matching,  at  80  cents  a yard ; or  1 yard  in  width,  with  10 
inches  allowance  for  matching,  at  $1  a yard.  Laying  the  strips  lengthwise, 
which  is  the  most  economical  ? 


PRACTICAL  MEASUREMENTS 


123 


THE  CIRCLE 

440.  A circle  is  a plane  figure  bounded  by  a curved  line  every  point 
of  which  is  equally  distant  from  a point  within  called  the  center. 

441.  The  circumference  of  a circle  is  its  boundary  line. 

442.  The  diameter  of  a circle  is  any  line  passing  through  the  center,  ending 
in  the  circumference. 

443.  The  radius  of  a circle  is  one-half  the  diameter,  or  any  line  drawn 
from  the  center  to  the  circumference. 

444.  Concentric  circles  are  cir- 
cles having  the  same  center. 

445.  To  find  the  circumference 
of  a circle. 

446.  Rule. — Multiply  the  diameter 
by  3.14-16. 

447.  To  find  the  diameter  of  a 
circle. 

448.  Rule. — Divide  the  circumference  by  3.1416. 

449.  To  find  the  area  of  a circle. 

450.  Rule. — Multiply  the  circumference  by  \ of  the  diameter,  or  the  diameter 
by  4 of  the  circumference. 


WRITTEN  PROBLEMS 

451.  1.  What  is  the  circumference  of  a wheel  whose  diameter  is  60  inches? 

3.  What  is  the  diameter  of  a circle  whose  circumference  is  628.32  rods? 

3.  A carriage  wheel  whose  diameter  is  3f  feet  made  1400  revolutions  in 
going  a certain  distance.  Find  the  distance. 

4 . What  is  the  area  of  a fish-pond  whose  diameter  is  25  ft.? 

5.  Find  the  cost  of  building  a wall  around  a circular  garden  containing 
250000  sq.  ft.,  at  $5.50  per  linear  yard. 

6.  The  side  of  a square  is  22  ft.  Find  the  diameter  of  a circle  equal  in  area 
to  the  area  of  the  square. 

7.  What  will  it  cost,  at  18  cents  a square  yard,  to  gravel  a walk  6 ft.  wide 
outside  a circular  plot  of  ground  whose  diameter  is  80  yards? 

8.  At  $2  a square  (100  sq.  ft.)  what  will  it  cost  to  paint  the  surface  of  a 
smoke-stack  31  ft.  6 in.  in  circumference  and  120  ft.  high  ? 

9.  The  radius  of  the  smaller  of  two  concentric  circles  is  10  feet,  and  that  of 
the  larger  circle  is  15  feet;  what  is  the  area  of  the  ring  between  the  two  circles? 


124 


PRACTICAL  MEASUREMENTS 


SOLID  MEASUREMENTS 

452.  Under  this  head  come  such  practical  measurements  as  masonry,  brick- 
work, excavations,  lumber,  contents  of  heavy  timbers,  wood  measure,  capacities 
of  tanks  and  bins. 

453.  A volume  has  length,  breadth,  and 
thickness  or  height. 

454.  A cube  is  a rectangular  solid,  the  six- 
faces  of  which  are  equal. 

455.  Students  should  note  carefully  the  fol- 
lowing facts : 

1.  27  cu.  ft.=l  cu.  yd.  or  load  ; 24f  cu.  ft.=a  perch  of  stone. 

2.  In  estimating  the  amount  of  material  required  for  a wall,  deductions  are  made  for  all  open- 
ings and,  in  a building,  for  the  corners,  the  latter  being  four  times  the  thickness  of  the  wall. 

3.  In  estimating  the  value  of  masons’  labor  on  walls,  no  allowance  is  made  for  corners  and  only- 
part  allowance  for  the  openings. 

J.  Because  of  the  fact  that  the  number  of  cubic  feet  in  a perch  varies  so  much  in  different  locali- 
ties, the  practise  of  estimating  by  the  perch  is  being  discontinued,  and  all  kinds  of  heavy  masonry, 
whether  brick,  stone  or  concrete,  is  now  quite  generally  estimated  by  the  cubic  yard. 

5.  Since  2*,  the  difference  between  27  cubic  feet  and  24f  cubic  feet,  is  TJT  of  the  latter  or  Tq  of 
the  former,  cubic  yards  may  be  changed  to  perches  by  adding  yU,  or  perches  may  be  reduced  to  cubic 
yards  by  subtracting  T\. 

6.  In  southeastern  Pennsylvania  and  adjacent  parts  of  New  Jersey,  it  is  customary,  in  the 
absence  of  a special  contract,  to  allow  only  22  cubic  feet  to  the  perch  in  measuring  walls  to  estimate 
either  labor  or  material 

456.  To  find  the  contents  of  walls  or  other  solids  in  cubic  feet,  cubic 
yards,  or  perches. 

Example.  — How 
many  cubic  feet  in  a stone 
wall  81  ft.  long,  22  in. 
thick  and  8 ft.  high  ? How 
unury  cubic  yards?  How 
many  perches? 


Note. — Only  like  dimen- 
sions should  be  multiplied,  hence, 
write  22  in.  as  \\  ft. 

Suggestion. — 24|  = -9^ 
which,  being  the  divisor,  is  in- 
verted in  the  solution,  and  the 
work  shortened  by  cancelation. 


457.  Rule. — Multiply  together 
the  three  dimensions— length,  width 
(or  thickness)  and  height  (or  depth) ; 
the  result  will  be  the  contents  in  cubic 
units  of  the  given  denomination. 


r/  x x fr 

/ 2 


/ tr  ir 


f / X 2 2 X r setts' 

/2  x xy  ' 

X / X 22  X X X £/ fX 

/2  X f f 


PRACTICAL  MEASUREMENTS 


125 


WRITTEN  PROBLEMS 

458.  1-  How  many  perches  of  stone  in  a wall  57  ft.  long,  7 ft.  high,  2J  ft. 
thick  ? What  will  be  the  cost  of  laying  at  75  cents  a perch  ? 

3.  How  many  perches  of  stone  in  a cellar  wall  18  in.  thick,  measuring  36  ft. 
by  24  ft.  outside,  8 ft.  deep  ? 

3.  What  will  it  cost  to  lay  a stone  wall  150  yards  long,  4 ft.  high,  20  in. 
thick,  at  95  cents  a perch  ? 

If.  What  will  it  cost  to  excavate  a reservoir  50  meters  square,  2 meters  deep, 
at  65  cents  a cubic  meter? 

5.  What  will  it  cost  to  cover  the  bottom  of  the  reservoir  in  preceding 
problem  with  concrete  1 inch  thick,  at  45  cents  a square  yard  ? 

6.  How  many  cubic  feet  of  earth  must  be  removed  in  digging  a ditch  120 
rods  long,  30  inches  wide,  3J  feet  deep  ? What  will  be  the  cost  of  the  work  at  55 
cents  a cubic  yard  ? 

7.  Find  the  cost  of  digging  a cellar  24  feet  wide,  54  feet  long  and  6 feet  deep 
at  35c.  a cubic  yard  for  f of  it,  and  45c.  a cubic  yard  for  the  remainder. 

8.  Find  the  cost  of  the  material  for  the  walls  of  a cellar  25  ft.  6 in.  long, 
18  ft.  9 in.  wide,  6 ft.  high  and  1J  ft.  thick  at  $3.75  a perch. 

9.  What  will  it  cost  to  construct  a retaining  wall  90  feet  long,  28  in.  thick, 
and  of  an  average  height  of  7 ft.,  at  $2.70  a perch  for  material  and  60  cents  a 
perch  for  labor  ? At  $3.60  a cubic  yard  for  both  ? 

10.  At  $3.25  a perch  for  the  material  and  $1.30  a perch  for  the  labor,  what 
will  it  cost  to  build  a stone  house  60  ft.  long,  42  ft.  6 in.  wide,  28  ft.  high,  the  walls 
being  16  inches  thick,  allowing  524  cu.  ft.  for  doors  and  windows? 

11.  A stone  wall  is  65  ft.  long,  2 ft.  thick  and  16J  ft.  high.  How  many 
perches  does  it  contain  ? How  many  cubic  yards  ? 

13.  The  foundation  of  a house  is  to  be  40  ft.  long  and  24  ft.  wide.  If  built 
20  inches  thick  and  ft.  high,  what  will  it  cost  at  $3  a perch  for  the  material 
(net  measure)  and  $2.50  a perch  for  the  mason  work  (outside  measure),  no  allow- 
ance being  made  for  openings? 

13.  Find  the  cost  of  digging  a cellar  108  ft.  long,  46  ft.  wide  and  10  ft.  deep 
at  44c.  a cubic  yard. 

Ilf.  What  will  it  cost  to  build  a foundation  wall  in  the  cellar  of  the  pre- 
ceding problem,  the  wall  to  be  2J  ft.  thick  and  12  ft.  high,  at  $3.75  a perch  for 
the  material,  and  $2.75  a perch  for  the  labor,  if  in  estimating  the  material  full 
allowance  is  made  for  2 openings  each  6 ft. X8  ft.  and  10  openings  each  3 ft.  X6  ft., 
and  f of  these  openings  is  deducted  in  estimating  the  masonry  ? 


126 


PRACTICAL  MEASUREMENTS 


15.  A foundation  wall  2 ft.  thick  and  8 ft.  high  for  a building  60  ft.  long  and 
28  ft.  6 in.  wide  contains  2 openings  for  doors,  each  7 ft.  by  4 ft.,  and  10  windows, 
each  6 ft.  by  3|  ft.  Find  the  cost  of  the  material  at  $3.20  a perch,  allowing  for 
all  the  openings;  also  find  the  cost  of  the  labor  at  $3.50  a perch,  allowing  for 
half  the  openings. 

16.  How  many  bricks  8 in.  X 4 in.  X 2 in.  will  be  required  for  a wall  60  feet 
long,  24  feet  high  and  three  bricks  thick? 

Note. — The  num- 
ber of  bricks  required 
for  a wall  is  usually 
found  by  calculating 
the  number  needed  for 
a course  and  multiply- 
ing by  the  number  of 
courses.  An  allowance 
is  made  for  mortar, 
usually  from  | of  an 
inch  to  £ of  an  inch,  to 
each  dimension. 

This  allowance 
need  not  be  considered 
in  working  these  prob- 
lems. 

17.  What  would  be  the  cost  of  bricks  required  in  preceding  problem  at 
$8.50  per  M ? 

18.  In  a brick-yard  there  is  a pile  of  common  bricks,  72  feet  8 inches  long, 
32  feet  4 inches  wide  and  12  feet  high.  What  are  the  bricks  worth  at  $8.50 
per  M ? 

19.  How  many  common  bricks  (8  in.  X 4 in.  X 2 in.)  are  required  for  the 
walls  of  a house  48  ft.  long,  30  feet  6 inches  wide,  25  feet  high  and  2 feet  thick, 
deducting  132  cubic  feet  for  windows  and  doors? 

20.  How  many  bricks  will  be  required  to  build  a bouse,  the  walls  of  which 
are  50  ft.  8 in.  long,  25  ft.  6 inches  wide,  42  ft.  high  and  2 ft.  thick,  allowing  21 
bricks  to  a cu.  ft.?  Allow  for  4 doors  8 ft.  high,  3 ft.  6 inches  wide,  and  10 
windows  7 ft.  long,  3 ft.  wide.  What  will  the  bricks  cost  at  $8.50  per  M? 

21.  A brick  house  measures  on  the  outside  40  ft.  9 in.  long,  34  feet  wide  and 
32  ft.  6 in.  high,  the  walls  are  24  inches  thick.  Making  an  allowance  for  12 
windows  6 ft.  3 in.  long,  3 ft.  wide,  and  6 doors  8 ft.  long,  4 ft.  wide,  what  would 
be  the  cost  of  the  bricks  at  $8  per  M,  allowing  21  bricks  to  the  cu.  ft.? 

22.  How  many  perches  of  stone  in  three  bridge  piers,  each  24  ft.  long,  40  ft. 
high,  12  ft.  thick  at  the  base  and  5 ft.  thick  at  the  top?  How  many  cubic  yards? 


PRACTICAL  MEASUREMENTS 


127 


LUMBER 

459.  An  amount  of  lumber  is  usually  given  in  board  feet,  except  in  the  case 
of  a few  rare  hardwoods,  such  as  black  ebony,  lignum  vitae,  German  black  walnut 
and  vermilion,  which  are  sold  by  the  pound  ; and  mouldings,  which  are  sold  by 
the  running  foot  at  so  much  per  C. 

460.  A board  foot  is  a square  foot  of  board  one  inch  thick  or  less. 

Note. — That  is,  in  lumber  dealers’  usage,  a foot  of  lumber  is  (or  is  the  equivalent  of)  a board 
twelve  inches  long,  twelve  inches  wide,  and  one  inch  thick  or  less — surface  alone  being  considered  ; but 
if  the  board  be  more  than  one  inch  thick,  a square  foot  of  it  makes  proportionately  more  than  a board 
foot.  Thus  if  11,  11,  or  2 inches  thick,  a square  foot  of  board  equals  one  and  one-fourth,  one  and  one- 
half,  or  two  board  feet ; and  so  on  with  greater  thicknesses. 

A mouIding=foot  is  1 inch  wide,  1 inch  thick  and  1 foot  long.  Mouldings  are  estimated  by 
the  size  of  the  pieces  from  which  they  are  cut,  as  shown  by  the  thick  edge  and  back  ; that  is,  if  the 
moulding  is  made  from  2 in.  X 3 in.  lumber,  every  foot  in  length  is  counted  6 moulding-feet. 

461.  To  find  the  amount  of  lumber. 

Example. — How  many  feet  of  lumber  (board  feet)  in  26  pieces,  3 inches 
thick,  4 inches  wide,  12  feet  long  ? 

Note. — In  a lumber  bill  this  is  written  26  pc.,  3X4  — 12. 


/ 2 X 4 X^3  X 2 & = 3 / 2 

Formula  (when  width  is  in  inches): 

Length  X Width  X Thickness  X Pieces  _ , „ 

— 2 — Board  Feet. 

Note. — The  operation  may  frequently  be  shortened  by  cancelation. 


WRITTEN  EXERCISE 

462.  1 ■ How  many  board  feet  in  200  inch  boards  10  in.  wide,  16  feet  long? 
How  many  board  feet  in  150  2-inch  planks,  9 inches  wide,  18  feet  long? 
3.  What  amount  of  lumber  in  80  joists  3X7 — 20? 

A How  many  feet  in  140  scantlings  3X4 — 18  ? 

5.  How  much  lumber  in  36  sills  6X8 — 30  ? 

6.  What  amount  of  lumber  in  36  plates  4X6 — 30? 

7.  Find  the  amount  of  lumber  in  180  pieces  2x4 — 16. 

8.  Find  the  amount  of  lumber  in  225  planks  2x9 — 14. 

,9.  How  many  feet  in  75  rafters  3X5 — 22? 

10.  How  many  feet  in  24  girders  3X10 — 28? 


128 


PRACTICAL  MEASUREMENTS 


11.  Find  amount  of  lumber  in  following  order  : 

Solution 


18  pc.  3X7—20 

18  pc.  3X7—20 

630 

36  “ 3X4—18 

36  “ 3X4—18 

648 

50  “ 2X10—12 

50  “ 2X10—12 

1000 

2278 

12.  Find  the  amount  of  lumber  in  the  following  bill: 

40  pc.  3X4—20 
12  “ 4X6—30 
8 “ 4X6—20 
60  “ 2X4—20 
140  “ 2X10—12 

13.  Find  board  feet  in  the  following  order: 

50  pc.  10X12—36 
120  “ 6X  8—30 
170  “ 4X  6—20 
2500  ft.  2 in.  plank 

14-  What  amount  of  lumber  in  the  following  bill? 

65  pc.  10X12—44 
150  “ 6X  8—36 
224  “ 4X  6—24 
880  “ 3X  7—36 
880  “ 2X  6—36 


15.  What  is  the  value  of  280  pc.  3x4 — 16  @ $14  per  M? 

Solution 


280X^X^X16^480  11. 
n 
1 


4.480 

14 

17.920 

44.80 

Result  $62,720 


16.  What  is  the  cost  of  the  following  bill  of  hemlock  at  $27.50  per  M? 

60  pc.  4 X 6 — 24 
75  “ 3 X 4—18 
120  “ 2X10—12 


17.  What  is  the  value  of  the  following  pine  lumber  at  $55.50  per  M ? 

20  pc.  14x20 — 45 
62  “ 12X16—40 


18.  What  is  the  cost  of  the  following  bill  of  oak  at  $56  per  M? 

50  pc.  4X6 — 20 
50  “ 3X4—10 
100  “ 2X6—10 
1200  ft.  boards 


PRACTICAL  MEASUREMENTS 


129 


WOOD  MEASURE 


463.  The  cord  is  the  unit  of  measure  for  rough  wood. 

464.  A cord  of  wood  is  a pile  8 feet  long,  4 feet  high  and  each  stick  4 feet 
long.  It  contains  128  cubic  feet. 

465.  A cord  foot  is  a portion  of  the  cord  1 foot  long. 

466.  Rule. — Multiply  together  the  length , height , and  width  ( each  expressed 
in  feet ) and  divide  by  128  to  get  cords. 

WRITTEN  PROBLEMS 

467.  1.  A pile  of  wood  is  84  ft.  long,  12  ft.  high,  and  the  sticks  are  6 ft.  long. 
What  is  it  worth  at  $3.25  a cord  ? 

2.  A train  of  18  cars,  each  44  ft.  long,  74  ft.  wide,  is  piled  with  wood  to  a 
height  of  7 ft.  Of  how  many  cords  does  the  load  consist? 

3.  A shed  32  ft.  long  and  14  ft.  wide  is  high  enough  to  hold  35  cords.  How 
much  will  it  cost  to  paint  the  outside  at  4J  cents  a square  foot? 

4~  A roof  projecting  18  in.  on  all  sides,  covers  a pile  of  wood  27  ft.  wide. 
The  boards  for  the  roof  cost  $47,564  at  $22.65  per  M.  What  is  the  value  of  the 
wood  piled  under  it  to  a height  of  15  ft.  8 in.,  at  $3.78  a cord? 

5.  A pile  of  wood  25  ft.  wide,  64  ft.  long  and  15  ft.  high,  contains  how  many 
cords  ? 

6.  What  is  the  value  of  a pile  of  wood  30  ft.  long,  8 ft.  wide  and  7J  ft.  high, 
at  $4.50  a cord  ? 

7.  From  a pile  of  pulp-wood  84  ft.  long,  8 ft.  wide  and  12  ft.  high,  48  cords 
were  sold.  What  was  the  length  of  the  pile  remaining? 


130 


PRACTICAL  MEASUREMENTS 


CAPACITIES 


468.  To  find  the  capacity  of  boxes,  bins,  or  tanks,  in  gallons  or 
bushels. 

469.  R ule. — Find  contents  in  cubic  inches  and  divide  by  number  of  cubic 
inches  in  a gallon  ( Liquid  231)  to  get  gallons  ; by  2150.1/.  cubic  inches  to  get  stricken 
bushels,  or  by  271/7.7  cubic  inches  to  get  heaped  bushels. 

WRITTEN  PROBLEMS 

470.  1.  A bin  7 ft.  6 in.  long,  4 ft.  9 in.  wide  and  5 ft.  6 in.  deep  is  filled 
with  wheat.  What  is  it  worth  at  974  cents  a bushel? 

2.  A rectangular  tank  is  8 ft  long,  5 ft.  wide  and  3 ft.  4 in.  deep.  How 
many  gallons  of  water  will  it  hold?  How  many  barrels? 

3.  If  the  tank  in  the  preceding  problem  were  converted  into  a bin,  how 
many  bushels  of  wheat  would  it  hold?  How  many  bushels  of  potatoes? 

1/.  How  many  bushels  of  wheat  will  a bin  6 feet  square  and  4 feet  deep 
contain  ? 

5.  What  is  the  capacity,  in  barrels,  of  a tank  8 feet  square,  10  feet  deep? 

6.  A farmer  has  a bin  24  ft.  long,  8 ft.  4 in.  wide  and  9 ft.  high.  How 
many  bushels  of  corn  in  the  ear  will  it  hold?  How  many  bushels  of  wheat  in 
it  when  it  is  filled  to  within  IS  inches  of  the  top  ? 

7.  A bin  15  ft.  long,  S ft.  4 in.  wide  and  6 ft.  3 in.  deep  was  filled  with 
apples,  from  which  enough  cider  was  pressed  to  fill  a tank  9 ft,  4 in.  long,  4 ft.  6 
in.  wide  and  3 ft.  deep.  If  the  apples  were  bought  at  374  cents  a bushel  and  the 
cider  was  sold  at  33J  cents  a gallon,  how  much  was  gained,  if  the  charge  for 
making  the  cider  was  75  cents  a barrel  ? 


PRACTICAL  MEASUREMENTS 


131 


GAUGING 

471.  Gauging  is  a process  of  finding  the  capacity  or  volume  of  tanks,  casks, 
barrels,  etc. 

472.  To  find  the  capacity,  in  gallons,  of  a round  tank  of  uniform 
diameter. 

473.  Rule. — Multiply  the  square  of  the  diameter  in  inches  by  the  height  or  depth 
of  the  tank  in  inches,  and  this  product  by  .0034- 

WRITTEN  PROBLEMS 

474.  1.  What  is  the  capacity  of  a tank  8 feet  4 inches  in  diameter  and  12 
feet  6 inches  deep  ? 

2.  Find  the  capacity  of  a stand-pipe  10  feet  in  diameter  and  180  feet  high. 

3.  An  oil  tank  5 ft.  8 in.  in  diameter  and  24  ft.  2 in.  long  (inside  measure- 
ment) is  filled  with  oil.  What  is  it  worth  at  6J  cents  a gallon? 

4 ■ A well  3 ft.  4 in.  in  diameter  has  12  feet  of  water  in  it.  How  many 
barrels  of  water  are  there  in  the  well? 

475.  To  find  the  capacity  of  a cask  or  a barrel  in  gallons. 


A cask  is  equivalent  to  a cylinder,  having  the 
same  length  and  a diameter  equal  to  the  mean  diam- 
eter of  the  cask. 


476.  Rule. — To  find  the  mean  diameter  of  a barrel  or  cask,  add  to  the  head 
diameter  § of  the  difference  between  the  head  and  the  bung  diameters,  expressed  in 
inches ; or,  if  the  staves  are  but  little  curved,  -f. 

Multiply  the  square  of  the  mean  diameter  by  the  length  ( both  expressed  in  inches), 
and  this  product  by  .0034  >'  the  result  will  be  the  capacity  in  gallons. 

WRITTEN  PROBLEMS 

477.  1.  How  many  gallons  in  a cask  whose  head  diameter  is  24  inches,  bung 
diameter  30  inches  and  its  length  34  inches  ? 

2.  What  are  the  contents  of  a cask  3 feet  6 in.  long,  the  head  diameter 
being  26  inches  and  the  bung  diameter  30  inches? 

3.  A merchant  received  ten  casks  of  New  Orleans  molasses,  having  the 
following  dimensions : head  diameter,  30  inches  ; bung  diameter,  38  inches  and 
length  42  inches,  at  an  average  cost  of  38  cents  per  gallon.  He  retailed  the  same 
at  12  cents  per  quart.  What  was  his  gain  on  the  entire  lot? 

4 . How  much  did  a farmer  realize  from  the  sale  of  7 barrels  of  vinegar  at 
12f  cents  a gallon,  if  the  staves  were  slightly  curved,  the  head  diameter  being 
2 ft.,  the  bung  diameter  2 ft.  3 in.  and  the  length  2 ft.  10  in.? 

5.  The  bung  diameter  of  a hogshead  is  45  inches,  its  head  diameter  40 
inches  and  its  length  55  inches  ; what  is  its  capacity  ? 

6.  How  many  gallons  in  a cask  of  slight  curvature,  3 ft.  6 in.  long,  the  head 
diameter  being  26  in.  and  the  bung  diameter  31  in.? 


132 


PRACTICAL  MEASUREMENTS 


GENERAL  REVIEW  PROBLEMS 

478.  1.  How  many  yards  will  it  take  to  make  a rug  4 yds.  wide  and  9 yds. 
long,  including  a border  18  inches  wide,  if  the  strips  in  the  body  of  the  rug  are 
made  of  27-inch  Brussels  and  9 in.  are  lost  on  each  strip  in  matching  ? 

2.  What  weight  of  water  is  contained  in  a tank  14  in.  X 9|  in.  X 8J  in.,  a 
b-inch  cube  of  water  weighing  125  oz.? 

3.  Find  the  cost  of  slating  a board  23  ft.  6 in.  long  and  2 ft.  8 in.  wide  at  9 
•cents  per  square  foot. 

4 Divide  J of  25.08J  miles  by  % and  change  to  proper  denominations. 

5.  Find  the  value  in  American  money  of  express  money  orders  amounting 
to  £17  9s.  6 d.  (Exchange  $4.8065.) 

6.  What  is  the  total  cost  of  24  tons  (2240  lb.)  of  wheat  at  90  cents  per 
bushel  and  140  centals  of  wheat  at  87J  cents  per  bushel? 

7.  What  is  the  depth  of  a tank  7 feet  long  and  4 feet  wide  that  will  hold  42 
barrels? 

8.  What  is  the  weight  of  shelled  corn  that  will  exactly  fill  a bin  8 ft.  long, 
■5  ft.  wide,  6 ft.  deep  ? 

9.  How  many  Philadelphia  bricks  8J  X 4|  X 2§  in  a pile  80  feet  X 6 feet 
X 12  feet?  What  is  their  value  at  $8  per  M ? 

10.  A tank  14  feet  long,  3 feet  8 inches  wide  and  11  feet  deep  is  filled  with 
petroleum.  What  is  its  value  at  15  cents  per  gallon? 

11.  If  48  gallons  of  wine  and  12  gallons  of  water  are  mixed,  how  much  wine 
is  there  in  4 qt.  1 pt.  of  the  mixture? 

12.  If  a miller  takes  § for  toll,  and  a bushel  of  wheat  produces  40  pounds  of 
flour,  how  many  bushels  must  be  taken  to  mill  to  obtain  10  barrels  of  flour  9 

13.  What  will  a tract  of  land  270  rods  long  and  495  yards  wide  cost  at  $624 
per  acre  ? 

111..  It  requires  24  men  30  days  to  do  a piece  of  work  After  they  have 
worked  8 days,  6 men  leave.  In  what  time  can  the  entire  work  be  completed? 

15.  The  sum  of  two  numbers  is  367  and  their  difference  is  11.  What  are 
the  numbers  ? 

16.  What  will  the  following  lumber  cost  at  $56.25  per  M? 

75  pc.  5 X 7 — 28 
75  “ 3 X 7 — 14 
150  “ 2 X 8 — 14 
300  “ 1 X 4 — 14 

17.  A man  wishing  to  build  a tank  in  his  barn  found  that  he  could  not 
safely  place  there  a weight  of  more  than  24  tons  of  water.  How  long  may  he 
make  a tank  4 feet  wide  and  4 feet  deep,  water  weighing  624  pounds  per  cubic 
foot  ? 

18.  What  will  it  cost  to  pump  the  water  out  of  a flooded  cellar,  30  feet  by 
18  feet,  the  water  being  4J  feet  deep,  at  8 cents  a hogshead? 

19.  How  many  bricks  8 in.  X 4 in.  X 2 in.  will  be  required  to  pave  a street 
14  miles  long  and  30  feet  wide,  if  the  bricks  are  laid  edgewise? 


PRACTICAL  MEASUREMENTS 


133 


20.  If  f of  A’s  money  equals  f of  B’s,  and  § of  B’s  equals  f of  C’s,  and 
together  they  have  $23300,  how  much  money  has  each? 

SI.  What  is  the  cost  of  550  inch  boards,  each  18  feet  long  and  4 inches  wide, 
at  $18J  per  M ? 

SS.  A farmer  sold  45  bushels  of  potatoes  and  55  bushels  of  rye  for  $55  50, 
receiving  10  cents  a bushel  more  for  the  rye  than  for  the  potatoes.  What  was 
the  price  of  each  per  bushel? 

S3.  Find  the  cost  of  carpeting  two  rooms,  each  21  feet  by  18  feet,  with  Brus- 
sels carpet  f yard  wide  at  $1.12J  per  yard  ; carpet  to  be  laid  lengthwise  and  1 
foot  allowed  to  each  strip  for  waste. 

Sj. £.  Divide  .06§  acres  by  1.66§,  and  multiply  the  quotient  by  9.3§. 

55.  A grain  elevator  is  40  feet  long,  18  feet  wide  and  25  feet  deep;  it  is  § 
full  of  wheat.  How  many  bushels  are  in  it?  What  is  the  weight  of  the  wheat 
in  it9 

56.  If  a bin  would  hold  1000  bushels  of  grain,  how  much  water  would  it 
contain  ? 

27.  What  is  gained  or  lost  by  buying  15  bushels  of  chestnuts  at  $5  per 
bushel,  dry  measure,  and  selling  them  at  25  cents  per  quart,  liquid  measure? 

28.  Bought  in  Paris,  France,  325.5  meters  of  silk  at  15.60  francs  per  meter. 
Find  cost  in  United  States  money. 

29.  If  the  driving  -wheels  of  a locomotive  are  10  feet  6 inches  in  circumfer- 
ence, and  average  8 revolutions  per  second,  how  long  will  it  require  to  go  76  mi. 
108  rd.  3 yd.? 

30.  A rectangular  field  140  rods  long  produces  1764  bushels  of  wheat  at  the 
rate  of  32  bushels  per  acre.  Find  its  width. 

31.  At  25  cents  per  square  foot,  find  the  cost  of  a walk  3 feet  wide  around  a 
grass  plat  81  feet  long  and  60  feet  wide;  also  the  cost  of  a drive  10  feet  wide 
around  the  walk  at  the  same  price  per  square  foot. 

32.  Bought  36500  tons  of  iron  at  £14  10s.  6 d.  per  ton.  What  is  its  value  in 
United  States  money  ? How  should  it  be  sold  per  ton  in  United  States  money  to 
gain  $2J  per  ton,  and  allow  $2J  per  ton  duty  ? 

33.  A bought  320  long  tons  of  hay  at  $15  per  ton  and  sold  it  at  $16.25  per 
short  ton.  Find  his  gain. 

31t-  A rectangular  iron  tank,  including  the  top,  is  made  of  2-inch  plates.  If 
the  inside  dimensions  of  the  tank  are  8 feet  long,  5 feet  wide  and  10  feet  deep,  how 
many  cubic  ft.  of  iron  in  the  material  ? 

35.  Find  the  weight  of  the  tank  in  preceding  problem,  and  its  contents 
when  filled  with  water,  if  iron  is  7 times  heavier  than  water. 

36.  How  many  feet  of  siding  would  be  required  for  a house  37  feet  6 inches 
long,  25  feet  wide  and  18  feet  high,  with  two  gables  each  12  feet  high,  adding  ^ 
for  lap  and  waste  ? 

37.  How  many  perches  of  stone  in  a wall  396  feet  long,  44  feet  wide  and  6.3 
feet  deep?  What  will  it  cost  to  build  the  wall  at  $3.25  per  perch? 

38.  When  wheat  is  quoted  at  $1.25  per  cental,  what  are  250  bushels  worth  ? 


134 


PRACTICAL  MEASUREMENTS 


39.  45  men  agreed  to  do  a piece  of  work,  but  25  of  them  did  not  come  and 
the  work  was  prolonged  5 days.  In  what  time  could  the  45  have  done  the  work  ? 

40.  What  should  be  paid  for  a piece  of  cloth  35 J yards  long  and  1J  yards 
wide,  if  $52.50  is  paid  for  a piece  of  the  same  quality  15f  yards  long  and 
yards  wide  ? 

41.  Find  the  cost  of  painting  the  walls,  floor  and  ceiling  of  a room  32  feet 
by  21  feet  and  15  feet  6 inches  high  at  19J  cents  per  square  yard. 

42.  Find  the  cost  of  the  following,  hemlock  being  worth  $40.50  per  M,  and 
pine  $24.50  per  M : 

80  pc.  Hemlock  6 X 8 — 32 
160  “ “ 4X6—24 

240  “ “ 2X10—24 

12000  ft.  Pine  boards 

4-5.  What  will  18  two-inch  planks  cost  at  $13J  per  M,  each  14  feet  long,  10 
inches  wide  at  one  end,  8 inches  at  the  other? 

44-  A grocer  purchased  eggs  at  20  cents  per  dozen  and  sold  them  at  the  rate 
of  6 for  19  cents.  How  many  must  he  sell  to  gain  $1.20? 

45.  A pharmacist  bought  210  pounds  of  drugs  at  72  cents  a pound,  avoir- 
dupois, and  sold  them  at  72  cents  a pound,  apothecaries’  weight.  Find  bis  gain. 

46.  Find  the  cost  of  a lot  176  feet  front  and  88  feet  deep,  at  $375.50 
per  acre. 

47.  A bin  15  feet  long,  6 feet  wide  and  9 feet  deep,  is  f full  of  wheat.  What 
is  the  value  at  90  cents  per  bushel  ? 

4.8.  How  many  board  feet  in  a board  26  feet  long,  15  inches  wide  and  4 
inch  thick  ? 

49.  What  will  be  the  cost  of  9.37125  miles  of  insulated  wire  at  3J  cents 
a foot  ? 

50.  If  a contractor  receives  $7425  for  grading  a roadbed  1|  miles  long,  what 
should  he  get  for  325.9  feet? 

51.  A man  bought  12  acres  of  land  at  $325  per  acre  and  laid  it  out  in  town 

lots  1 chain  50  links  long  by  1J  chains  deep.  He  sold  the  lots  at  S825  each. 

What  was  his  gain  ? 

52.  Certain  city  lots  are  24  feet  by  100  feet.  How  many  such  lots  in  an  acre  ? 

53.  At  2J  cents  a square  inch,  what  will  it  cost  to  bronze  a cube  the  edge  of 
which  is  3J  feet  ? 

54 • A farmer  sowed  3 bu.  1 pk.  1 qt.  of  seed  and  harvested  from  it  87  bu.  2 
pk.  and  3 qt.  How  many  bushels  did  he  harvest  from  a bushel  of  seed  ? 

55.  What  would  it  cost,  at  16  cents  per  square  yard,  to  paint  the  outside  of 
a house  56  feet  by  24  feet  by  23  feet  6 inches  in  height  ? 

56.  A can  plow  -f  of  a field  in  3 days  ; B f of  it  in  6 days.  How  long  will 

it  require  A and  B working  together  to  plow  it? 

57.  How  many  meters  in  a mile?  How  many  yards  in  565.25  meters? 

58.  If  a five-cent  piece  is  of  a meter  in  diameter,  bow  many  of  them  will 
extend  a yard  ? 


PRACTICAL  MEASUREMENTS 


135 


59.  How  many  rods  of  fence  will  be  required  to  fence  a railway  f of  a mile' 
long?  How  many  posts,  each  6 feet  apart?  How  many  feet  of  wire,  if  the  fence 
is  5 wires  high  ? 

60.  Reduce  § of  an  ounce  avoirdupois  to  the  decimal  of  a ton. 

61.  If  a lady  weighs  125  pounds  avoirdupois,  what  would  she  weigh  Troy? 

63.  The  height  of  a flight  of  stairs  is  18  feet.  How  many  steps,  each  8 

inches  high  ? How  many  yards  of  carpet  would  be  required  if  the  tread  is  ten 
inches,  allowing  1 yard  for  waste? 

63.  Divide  .0974  of  a mile  by  2.5,  and  multiply  the  quotient  by  .03J.  Change 
the  results  to  proper  denominations. 

64..  How  many  boards,  each  18  feet  long  and  6 inches  wide,  will  be  required 
to  make  a cubical  box  6 feet  long  ? 

65.  Find  the  cost  of  the  masonry  of  a building  at  $7.25  per  perch,  the 
dimensions  of  which  are  31  feet  6 inches  by  23  feet  6 inches  and  7 feet  high,  18 
inches  thick. 

66.  What  will  be  the  thickness  of  a piece  of  timber  50  feet  long,  10  inches 
wide,  to  contain  240  board  feet  ? 

67.  If  f of  $17g-  will  buy  oh  dozen  oranges,  how  many  dozens  can  you  buy 

for*®? 

b 

68.  What  will  it  cost,  at  $16  per  M,  to  erect  a tight  board  fence  6 feet  high 
around  a lot  60  yards  long  and  76  feet  wide  ? 

69.  Multiply  193.875  miles  by  .083,  and  divide  the  result  by  .055. 

70.  Find  the  cost  at  $90  per  acre  of  a field  15  chains  75  links  long  and 
9 chains  25  links  wide. 

71.  A paper  manufacturer  paid  $357  at  $8.50  a cord  for  a pile  of  pulp-wood 
32  feet  long,  5J  feet  wide.  How  high  was  it? 

73.  If  8 men  or  12  boys  can  do  a piece  of  work  in  10  days,  how  long  will  it 
require  8 men  and  15  boys  to  do  the  work  ? 

73.  If  a blackboard  21  feet  long  contains  9 sq.  yd.  6 sq.  ft.,  how  wide  is  it  ? 

74-  Add  .0|  sq.  yd.,  .03J  sq.  ft.  and  .18  sq.  in.  and  multiply  the  sum 
by  12.31. 

75.  Which  will  be  the  cheaper  and  how  much,  to  lay  carpet  27  inches  wide 
lengthwise  or  crosswise  in  a room  26  feet  long  and  21  feet  wide,  at  $1.25  per 
yard,  allowing  one  foot  to  each  strip  for  waste? 

76.  A bin  9 feet  long,  5 feet  wide,  contains  540  bushels.  Allowing  1 \ cu.  ft. 
4o  a bushel,  bow  deep  is  it? 

77.  Allowing  21  bricks  to  a cubic  foot,  how  many  bricks  would  be  required 
for  the  main  walls  of  a building  40  feet  long,  28  feet  9 inches  wide,  24  feet  6 inches 
high  and  24  inches  thick  ? What  would  be  the  cost  of  the  bricks  at  $10.50 
per  M ? 

78.  Divide  155  inches  by  .0625  and  change  to  higher  denominations. 

79.  If  8 barrels  of  potatoes  and  9 barrels  of  apples  are  worth  $30,  and  16 
barrels  of  potatoes  and  12  barrels  of  apples  are  worth  $48,  what  is  the  price  of 
one  barrel  of  each? 


136 


PRACTICAL  MEASUREMENTS 


80.  How  many  bundles  of  laths  will  be  required  for  the  walls  and  ceiling  of 
a room  28  feet  long,  15  feet  6 inches  wide  and  10  feet  high,  each  bundle  being 
estimated  to  cover  5 sq.  yd.? 

81.  A case  of  cranberries  containing  3T3g-  bushels  was  bought  at  $3  per 
bushel  and  retailed  at  10  cents  per  quart,  liquid  measure.  How  much  was 
gained  thereby  ? 

2 3 

82.  If  3 of  a property  cost  $700,  what  should  be  given  for  t of  a property 

4 % 

at  the  same  rate? 

83.  Find  the  cost  of  the  following  bill  of  lumber  : 

300  inch  boards,  14  ft.  long,  8 in.  wide,  at  $32.50  per  M. 

50  joists  12  ft.  long,  4 by  5 in.  “ 28  00  “ “ 

60  planks,  20  ft.  long,  10  by  2J  in.  “ 9.50  “ C 

84-  A grain  dealer  paid  $118  for  22  barrels  of  flour,  giving  $6  for  the  first 
grade  and  $4  for  the  second  grade.  How  many  barrels  did  he  purchase  of  each  ? 

85.  At  $.56  per  load,  what  will  the  excavation  of  a cellar  cost,  the  dimen- 
sions of  which  are  29  feet  6 inches  by  22  feet  8 inches  and  6.3  feet  deep? 

86.  How  many  centals  of  grain  will  a rectangular  box  9 feet  long,  4 feet  2 
inches  wide  and  18  inches  deep  hold,  allowing  60  pounds  to  a bushel  ? How 
many  gallons  of  water? 

87.  From  £17  8.s.  6d.  deduct  .05  of  itself. 

88.  How  many  sheets  of  tin  20  inches  by  14  inches  will  cover  a roof  60  feet 
long  and  23  feet  from  eaves  to  ridge? 

89.  A railroad  passes  through  1^  miles  of  A’s  farm.  If  the  road  is  60  feet 
wide,  what  is  the  cost  of  the  right  of  way  at  $66  per  acre? 

90.  If  45  T.  15  cwt.  of  coal  are  worth  $222.25,  what  is  the  value  of  23  T. 
12  cwt.? 

91.  The  floor  of  a pavilion  87  jrards  long  and  26  feet  wide  is  2J  inch  oak. 
What  is  it  worth  at  $37J  per  M? 

92.  Find  weight  in  long  tons  of  the  ice  64  inches  thick  on  a pond,  the 
dimensions  of  which  are  180  yards  by  36  feet,  if  water  expands  jig-  of  its  volume 
in  freezing  and  a cubic  foot  of  water  weighs  1000  ounces. 

93.  If  8 barrels  of  flour  cost  £15  8s.  6d.,  what  will  5 barrels  cost  at  the  same 
rate?  (Result  in  English  and  United  States  money.) 

94-  Find  the  cost,  at  8 cents  a pound,  of  lining  with  zinc  a cistern  (including 
the  top)  9 feet  6 inches  by  4 feet  3 inches  and  9 feet  deep,  if  for  every  5 sq.  ft. 
12  pounds  are  required. 

95.  How  many  half  acre  lots  can  be  laid  out  in  a tract  of  land  containing 
105  A.  30  sq.  rd  ? 

96.  At  $15  an  acre,  what  is  the  cost  of  a tract  of  land  34  miles  square? 

97.  How  many  feet  of  inch  boards  will  be  required  to  build  a fence  around 
a lot  35  rods  long  and  22  rods  wide,  if  the  fence  is  made  6 feet  high?  What  will 
be  the  cost  of  the  boards  at  $12.50  per  M ? 


PRACTICAL  MEASUREMENTS 


137 


APPROXIMATE  . MEASUREMENTS 

479.  For  practical  purposes  the  following  rules  approximate  accuracy. 
They  will  be  found  convenient  for  use  by  farmers  and  others  in  cases  in  which 
only  approximately  accurate  results  are  required. 

GRAIN 

480.  To  find  the  number  of  stricken  bushels  of  grain  in  a bin  or  box. 

Note. — The  relation  of  2150.4  eu.  in.  to  1728  cn.  in.  is,  approximately,  as  5 to  4 ; orljcu.  ft.  are 
nearly  a bushel  ; so  in  practical  work  the  number  of  eu.  ft.  diminished  by  1 will  give  an  equivalent  in 
bushels  and  the  bushels  increased  by  1 will  give  an  equivalent  in  cu.  ft. 

481.  Rule. — Multiply  together  the  three  dimensions  expressed  in  feet,  and  their 
product  by  4,  or  .8. 

Note  1. — For  greater  accuracy  add  1 bushel  for  every  100  cu.  ft. 

Note  2. — If  the  bin  or  box  is  only  partly  filled  use  the  depth  of  the  grain  for  the  third 
dimension. 

482.  To  find  the  number  of  heaped  bushels  in  a box  or  bin. 

483.  Rule. — Find  the  contents  in  cubic  feet  and  multiply  by  .63. 

484.  To  find  the  number  of  bushels  of  well-seasoned  corn  in  the  ear, 
in  a bin  or  crib. 

485.  Rule. — Multiply  the  cubic  feet  by  and  divide  by  9,  (if  the  corn  be  of 
good  quality);  if  of  inferior  quality,  simply  multiply  the  cubic  feet  by  .Jf. 

Note  1. — The  above  rule  is  based  on  the  fact  that  2J  cu.  ft.  of  good  corn  in  the  ear,  or  2J  cu.  ft. 
of  inferior  corn,  will  yield  1 bu.  of  shelled  corn — 56  lbs. 

Note  2.— If  the  crib  be  a flaring  one — wider  at  the  top  than  at  the  bottom — take  for  the  third 
dimension,  the  width  at  half  the  height  to  which  it  is  filled  with  corn,  the  corn  being  leveled  before 
measuring. 


HAY 

486.  Owing  to  the  variety  of  conditions  affecting  the  kind,  it  is  difficult  to 
make  an  accurate  estimate  of  the  quantity  of  hay  in  a given  space  or  bulk. 
Under  ordinary  conditions,  however,  the  following  facts  will  give  a fair  approxi- 
mate : A ton  of  hay  unpressed  in  a load  or  loft  is  540  cu.  ft.;  in  a common 
covered  hay  barn  or  in  a low  stack,  405  cu.  ft.;  if  timothy  hay,  in  mow  com- 
pressed with  grain  or  in  butts  of  large  stacks,  324  cu.ft.;  in  well-settled  large 
stack  or  mow,  450  cu.  ft.;  if  clover  hay,  550  cu.  ft. 

487.  Rule. — Find  contents  in  cu.  ft.  and  divide  by  the  number  of  cubic  feet  in 
kind  of  hay  required. 


COAL 

488.  Lehigh  white  ash  coal,  egg  size,  34J  cubic  feet  in  a ton;  Schuylkill 
white  ash  coal,  35  cubic  feet ; red  ash  stove  coal,  36  cubic  feet. 

489.  Rule. — Find  contents  in  cubic  feet  and  divide  by  the  number  of  cubic  feet 
for  a ton. 


138 


PRACTICAL  MEASUREMENTS 


LIQUIDS 


490.  To  find  approximately  the  capacity  in  gallons  or  barrels  of  a 
rectangular  tank  or  cistern. 

491.  Rule. — Multiply  the  contents  in  cubic  feet  by  7\  for  gallons,  or  divide  the 
cubic  feet  by  4-f  {4--®)  for  barrels. 

Note. — For  greater  accuracy  in  gallons,  deduct  1 gallon  for  every  50  cubic  feet. 

492.  To  find  the  approximate  capacity  of  a round  tank,  well,  cistern, 
etc.,  the  dimensions  of  which  are  given  in  feet. 

493.  R ule. — Multiply  the  square  of  the  diameter  in  feet  by  the  depth  in  feet  and 
this  product  by  5%.  The  result  will  be  the  capacity  in  gallons. 

Note — In  a flaring  or  sloping  tank  (one  smaller  at  the  top  than  at  the  bottom),  the  average 
or  mean  diameter  is,  approximately,  l the  sum  of  the  top  and  bottom  diameters. 

BRICKS  AND  SHINGLES,  OR  SLATES 

494.  To  find  the  approximate  number  of  bricks  required  for  a wall. 

Note. — 22  common  bricks  laid  in  mortar  equal  1 cu.  ft. 

495.  Rule. — Multiply  the  number  of  cubic  feet  in  the  wall  by  22. 

Note  1.— The  dimensions  of  common  bricks  are  8 in.  X 4 in.  X 2 in.;  of  Philadelphia  and 
Baltimore  bricks,  81  X 4J  X 2§  in.;  of  North  River  bricks,  8 X 34  X 21  in. ; of  Milwaukee  bricks, 
8J  X 41 X 2|  in. ; and  of  Maine  bricks,  71  X 3f  X 2f  in. 

Note  2. — To  find  the  number  of  fancy  bricks  for  outside  courses  of  walls,  multiply  the  square 
feet  of  surface  by  the  number  of  bricks  required  for  1 sq.  ft.  of  surface,  which  number  is  obtained  by 
dividing  144  by  the  exposed  surface  of  1 brick,  making  proper  allowance  for  mortar.  The  number  of 
fancy  bricks  multiplied  by  the  number  of  bricks  in  thickness  back  of  them  will  give  the  approximate 
number  of  common  bricks  required  for  the  rest  of  the  wall. 

496.  To  find  the  approximate  number  of  shingles  or  slates  for  a roof. 

Note  1. — If  laid  5 inches  to  the  weather  it  requires  71  shingles  to  make  a square  foot  of  roof,  or 
720  to  the  square  of  100  sq.  ft. ; if  laid  4 inches  to  the  weather,  it  requires  8 to  the  square  foot,  or  800  to 
the  square.  To  make  proper  allowance  for  waste,  builders  usually  estimate  800  to  1000  shingles  (or 
about  4 bundles)  to  every  100  square  feet  of  roof. 

Note  2. — Roofing-slate  is  sold  by  the  square  (100  sq.  ft. ),  the  price  depending  on  the  color  and 
the  quality,  as  well  as  on  the  number  of  pieces  required.  Sizes  vary  from  6"  X 12//  to  14//  X 24" — the 
length  always  being  in  even  inches,  but  the  width  being  in  either  even  or  odd  inches.  The  number  of 
pieces  to  the  square  depends,  of  course,  on  the  exposure,  which  dealers  usually  reckon  as  4 of  the 
remainder  after  deducting  3 from  the  length  of  the  slate.  Thus  the  exposure  of  a slate  18  in.  long  is 


PRACTICAL  MEASUREMENTS 


139 


LOOS 

497.  In  estimating  the  amount  of  lumber  in  square-edged  inch-boards  that 
can  be  sawed  from  a round  log,  lumbermen  and  others  make  use  of  the  follow- 
ing rules : 

498.  Doyle’s  Rule. — From  the  diameter  in  inches,  subtract  ^ ; the  square  of  the 
remainder  will  be  the  number  of  board  feet  yielded  by  a log  16  feet  long.  Or,  from 
the  diameter  in  inches,  subtract  f,  square  f of  the  remainder  and  multiply  the  product 
by  the  length  in  feet. 

499.  Ropp’s  Rule. — From  the  square  of  the  diameter  in  inches,  subtract  60, 
multiply  the  remainder  by  J the  length  in  feet,  and  point  off  the  right-hand  figure  of 
the  product. 

500.  Two  -thirds’  Rule. — From  the  diameter  deduct  } for  saw  kerf  (cut)  and 
slab  ; square  the  remainder,  multiply  by  the  length  of  the  log,  and  divide  this  product 
by  12 

501.  Three-fourths’  Rule. — Same 
as  preceding  rule,  except  that  J is  deducted 
for  saw  kerf  and  slab. 

Example. — How  many  square  feet  of 
inch  boards  can  be  cut  from  a log  24  in. 
in  diameter  and  20  ft.  long  ? 


SOLUTIONS 


Doyle’s  Rule 

24  — 4 = 20 
i of  20  = 5 
5 X 5 X 20  = 500 
Result  500  ft. 


Ropp’s  Rule 

24  X 24  = 576 
576  — 60  — 516 
516  X 10  = 516.0 
Result  516  ft. 


Two-thirds’  Rule 
24  — 8 =16 
16  X 16  = 256 
256  X 20  = 5120 
5120  = 12  = 4261- 
Result  426  ft. 


Three-fourths’  Rule 
24  — 6 =18 
18  X 18  = 324 
324  X 20  = 6480 
6480  12  = 540 

Result  540  ft. 


Note  1. — The  term  “ diameter  ” is  understood  to  meau  the  diameter  inside  the  bark  at  the  top 
or  smaller  end  of  the  log. 

Note  2. — No  rule  can  be  relied  upon  for  absolute  accuracy  under  all  conditions.  Doyle’s 
rule,  though  in  general  use,  is  too  favorable  to  the  buyer  of  small  logs  and  to  the  seller  of  large  ones. 
“The  Woodman’s  Handbook,”  issued  by  the  Bureau  of  Forestry,  Department  of  Agriculture,  Wash- 
ington, D.  C.,  gives  a complete  list  of  rules. 


WRITTEN  PROBLEMS 

502.  1.  How  many  bushels  of  wheat  or  other  small  grain  can  be  stored  in 
a granary  24  ft.  6 in.  long,  7 ft.  wide  and  6 feet  high? 

2.  How  many  bushels  of  apples,  potatoes,  or  turnips  can  be  put  into  a bin 
10  ft.  6 in.  long,  4 ft.  wide,  filled  to  a depth  of  3 feet.  9 in.? 

3.  A box  car  28  ft.  6 in.  long  and  8 ft.  wide  (inside  measurement)  is  filled 
with  lime  to  a depth  of  4 ft.  3 in.  How  many  bushels  does  it  contain? 

Jf.  A corn  crib  30  ft.  long,  9 ft.  high,  5 ft.  wide  at  the  bottom  and  8 ft.  wide 
at  the  top  is  filled  with  corn  of  good  quality.  How  many  bushels,  approximately, 
are  there  in  the  crib,  and  what  should  be  its  weight  when  shelled? 


140 


PRACTICAL  MEASUREMENTS 


5.  How  many  tons,  approximately,  of  clover  hay  are  there  in  a stack  20  ft. 
long,  15  ft.  wide  and  12  ft.  high,  and  what  is  the  hay  worth  at  $15  per  ton? 

6.  How  many  tons  of  Lehigh  stove  coal  can  be  put  into  a bin  14  ft.  long, 
10  ft.  wide  and  8 ft.  deep?  How  many  tons  of  Schuylkill  white  ash  stove  coal? 
How  many  tons  of  red  ash  stove  coal  ? 

7.  A dealer  has  a bin  filled  with  Schuylkill  white  ash  stove  coal  that  cost 
him  $405,  at  $4.50  per  ton.  The  bin  is  25  ft.  long  and  15  ft.  wide.  What  is  the 
depth  of  the  coal  ? 

8.  A farmer  sold  the  timothy  hay  in  a mow  30  ft.  long,  18  ft.  wide  and  11 
ft.  high,  at  $10.25  a ton  of  324  cu.  ft.  How  much  did  he  receive  for  it? 

9.  A contractor  paid  $150  for  a stack  of  hay.  It  was  30  ft.  long,  10  ft.  wide 
and  12  ft.  high.  What  was  the  price  of  the  hay  per  ton,  if  450  cubic  feet  were 
taken  as  the  equivalent  of  a ton  ? 

10.  A rectangular  tank  8 ft.  long,  3J  ft.  wide  and  2J  ft.  deep  will  contain  how 
many  gallons  when  filled? 

11.  A round  tank  4 ft.  8 in.  in  diameter  and  9 ft.  long  will  hold  how  many 
gallons  ? How  many  barrels  ? 

13.  A well  3 ft.  6 in.  in  diameter  and  30  ft.  deep  is  filled  with  water  to  within 
14  ft.  of  the  top.  How  many  hogsheads  of  water  in  it,  approximately? 

13.  Find  the  capacity  in  barrels  of  a tank  9 feet,  4 inches  in  diameter  at  the 
top,  11  feet  8 inches  at  the  bottom  and  18  feet  4 inches  deep. 

Ilf..  Find  the  contents  in  board  feet  of  two  logs  each  18  in.  in  diameter  at 
the  top  and  24  ft.  long.  (Results  by  different  rules.) 

15.  How  many  bricks  will  it  take  to  build  a wall  7 ft.  high,  17  inches  (or  4 
bricks)  thick,  on  three  sides  of  a garden  50  ft.  square,  allowing  for  a gate  6 feet 
wide  ? 

16.  How  many  fancy  bricks,  8 in.  X 2 in.  on  the  face,  will  it  take  for  the 
outside  course  of  a building,  45  ft.  long  and  28  ft.  wide,  the  walls  being  21  ft.  6 
in.  high  from  the  foundation  to  the  cornice,  deducting  4 doorways,  each  5 ft.  by 
10  ft.  and  16  windows,  each  4J  ft.  by  8 ft.?  What  will  they  cost  at  $27.50 
per  M ? 

17.  If  the  outside  walls  in  the  preceding  problem  are  5 bricks  thick,  and 
18  in.  all  around  the  top  of  them  is  made  entirely  of  common  bricks,  besides 
a total  length  of  150  ft.  of  3-brick  partition  walls  in  the  building,  averaging 
20  ft.  high,  and  80  ft.  of  2-brick  partitions  10  ft.  high,  how  many  common 
bricks  will  be  required?  What  will  they  cost  at  $12.50  per  M? 

18.  A farmer  hauled  to  a sawmill  20  straight,  smooth  logs  of  the  following 
dimensions:  2 logs  48  in.  in  diameter,  12  ft.  long;  2 logs  36  in.  diameter,  14  ft. 
long  ; 2 logs  30  in.  in  diameter,  16  ft.  long  ; 3 logs  24  in.  in  diameter,  IS  ft.  long  ; 
3 logs,  18  in.  in  diameter,  20  ft.  long  ; 2 logs,  15  in.  in  diameter,  22  ft.  long  ; 2 logs, 
12  in.  in  diameter,  24  ft.  long ; 4 logs  10  in.  in  diameter,  28  ft.  long.  What  did 
it  cost  him  to  have  these  logs  sawed  into  lumber  at  $6  per  thousand  board  feet  ? 
(Results  by  different  rules.) 


PERCENTAGE 


503.  Percentage  is  a method  of  computation  by  hundredths. 

504.  Per  cent,  is  an  abbreviation  of  the  Latin  phrase  per  centum,  and  means 
by  the  hundred.  The  sign  for  per  cent,  is  %. 

Note. — 10  % means  10  of  every  hundred,  or  ten  hundredths  (bfo)- 

505.  There  are  three  principal  elements  considered  in  percentage.  The 
Base,  the  Rate  and  the  Percentage. 

506.  The  Base  is  the  number  of  which  a number  of  hundredths  are  taken. 

507.  The  Rate  is  the  number  of  hundredths. 

508.  The  Percentage  is  the  result  obtained  by  taking  a certain  number  of 
hundredths  of  the  base. 


509.  In  computing  percentage,  it  is  frequently  more  convenient  to  use  the 
equivalent  common  fraction  than  the  decimal.  The  following  equivalents 
should  be  memorized  by  the  student: 


50% 

1 

2 * 

14*% 

i 

7‘ 

10% 

i 

— io- 

22*% 

9 

40- 

33*  % 

x 

— 3- 

25% 

1 

4- 

15% 

3 

— 2 0- 

20% 

i 

5- 

35% 

7 

2 0- 

66*  % 

2 

3- 

12*% 

1 

8* 

7J% 

3 

40- 

30% 

3 

— io- 

45% 

9 

2 0- 

16-1% 

1 

6’ 

75% 

3 

4* 

2i% 

1 

4 0- 

40% 

2. 

5 ‘ 

1 

14- 

83*% 

5 

6- 

37i% 

3 

8* 

13*% 

2 

15- 

60% 

3. 

5 ‘ 

22*% 

2 

9- 

11*% 

1 

9 • 

62i% 

5 

8* 

3*% 

1 

30- 

70% 

7 

— io- 

44*% 

4 

9- 

9*% 

1 

1 1 

87i% 

= i* 

18f% 

3 

16- 

80% 

4 

5 " 

28*% 

2 

7 ‘ 

8*% 

1 

12 

5% 

1 

— 2 0* 

17*% 

7 

40- 

90% 

9 

— io- 

6f% 

_U_ 

1 5 ‘ 

6*  % 

1 

— TS 

General  Percentage  Formula 


^ 0^  0j  C|aAK/vv  *uxaA- 

-vwbu XKbsJ,  av  cG/cAjmaAo 
l<x^u/v oAG 

Avllwo^'  byiMjj 
(GiW,,  /0 
mj ib 

SjA/O-AA'  /C<hAfc 
Lctoy,  fpv  cLwG|' 

A/v\^MAA/o/U/<yiy 

(hyiM/du/rub,  tfo  aAAiAAmT/ 
aTc/. 


V 


141 


142 


PERCENTAGE 


510.  To  find  the  percentage,  the  base  and  rate  being  given. 

Formula. — Base  X Rate  (expressed  decimally)=Percentage. 

Example. — A has  a farm  containing  375  acres ; 60%  of  it  is  in  wheat.  How 
many  acres  are  in  wheat  ? 

375  Base 
.60  Rate 

225.00  Percentage 

511.  Rule. — To  find  the  'percentage,  multiply  the  base  by  the  rate  expressed 
decimally.  Or,  take  the  equivalent  fractional  part. 

MENTAL  PROBLEMS 

512.  1.  A man  paid  $120  for  a horse  and  sold  it  at  a gain  of  10  per  cent. 
What  was  the  amount  of  gain  ? 

Solution. — A gain  of  10  per  cent,  is  a gain  of  ^ of  the  cost,  which  is  $12. 

2.  A milliner  sold  a hat  costing  $8  at  a gain  of  25  per  cent. ; what  does  she 

gain  ? 

3.  What  is  the  loss  on  a cow  bought  at  $40  and  sold  at  a loss  of  5 per  cent.  ? 

A A merchant  having  a stock  of  75  barrels  of  flour  sold  20  per  cent,  of 

them  ; how  many  barrels  were  left? 

5.  What  is  the  difference  between  20  per  cent,  of  $60  and  25  per  cent, 
of  $48  ? 

6.  On  a house  costing  $5000,  repairs  to  the  amount  of  5 per  cent,  are  made 
yearly;  what  is  the  amount  spent  for  repairs? 

7.  On  an  importation  of  watches  invoiced  at  $40  each,  a duty  of  20  per 
cent,  is  charged  ; how  much  per  watch  is  added  to  the  cost  for  duty? 

8.  On  a debt  of  $500,  monthly  instalments  of  10  per  cent,  are  promised 
until  settled;  how  much  must  be  paid  each  month? 

9.  There  must  be  added  for  waste  12J%  in  matching  carpet;  how  much 
must  be  added  for  a room  requiring  SO  yards  ? 

10.  I buy  a bill  of  goods  of  $360,  agreeing  to  pay  in  cash,  33J  per  cent.; 
what  is  the  amount  of  cash  to  pay? 

11.  I own  a one-half  interest  in  a business  enterprise  and  sell  33J  per  cent, 
of  my  interest ; what  part  of  the  business  do  I own  after  the  sale  ? 

12.  If  cloth,  when  sponged,  shrinks  5 per  cent.,  what  will  be  the  loss  from 
this  cause  in  sponging  40  yards? 

13.  An  employer  withholds  20  per  cent,  of  the  wages  of  his  employees  as  a 
guarantee.  What  would  be  held  from  the  wages  of  a person  receiving  $60  a 
month  ? 

Ilf..  The  duty  on  imported  tobacco  is  15  per  cent.;  what  amount  of  duty 
would  an  importer  have  to  pay  on  an  invoice  of  $600  ? 

15.  A merchant  reckons  his  yearly  loss  on  furniture  by  depreciation  at  121 
per  cent.;  if  the  last  inventory  showed  $1200,  what  must  the  new  one,  taken  at 
the  end  of  the  year,  show  ? 

16.  On  an  invoice  of  merchandise  weighing  600  pounds,  31  per  cent,  is 
allowed  for  packing  ; what  is  the  net  weight  of  the  goods  ? 


PERCENTAGE 


143 


WRITTEN  EXERCISE 


513.  1.  Find  2%  of  12694.  6. 

2.  3%  of  $289.63.  7. 

3.  7 % of  $2439.22.  8. 

Ip.  14%  of  $7896.  9. 

5.  19f%  of  $4293.85.  10. 


What  is  117%  of  32648? 
43  % of  $874.75? 

138%  of  $2550.80? 

77%  of  7777? 

29%  of  $32560? 


11. 

12. 

13. 

H- 

15. 

16. 
17. 


What  is  6J%  of  $3000.18? 

What  is  8J%  of  16  lb.  10  oz.  sugar? 
What  is  6 % of  £25  12s.? 

What  is  12i%  of  $43218.50  ? 

What  is  16f%  of  200  acres  15  sq.  rods  ? 
What  is  13^-%  of  $2500  ? 

What  is  44%  of  $37500? 


18.  What  is  2i%  of  3500000? 


WRITTEN  PROBLEMS 

514.  1.  Goods  costing  $5730  were  sold  at  a gain  of  30%  ; how  much  was 
the  gain  ? 

2.  A horse  costing  $175  was  sold  at  a loss  of  6%  ; what  was  the  loss? 

3.  A merchant  pays  his  debts  at  the  rate  of  35  cents  on  the  dollar  ; how 
much  does  a creditor  receive  whose  account  is  $5270? 

Ip.  On  a debt  of  $175.75,  15%  was  thrown  off  for  cash  ; what  amount  of  cash 
was  received  ? 

5.  A broker  bought  bonds  whose  face  value  was  $5500  and  received  J%  for 
buying ; how  much  did  he  receive  ? 

6.  The  marked  price  of  an  article  of  furniture  was  $300,  but  in  a reduction 
sale  the  price  was  marked  down  16§ % ; what  was  the  reduced  price? 

7.  A clerk  was  directed  to  mark  goods  that  cost  $50,  35%  above  cost;  what 
was  the  marked  price? 

8.  Goods  marked  20%  above  their  cost  price  of  $10  were  sold  5%  below  the 
marked  price  ; what  was  received  for  them  ? 

9.  A merchant  had  marked  his  stock  40  % above  cost.  He  opens  a reduction 
sale  and  marks  down  the  goods  50%  ; what  part  of  the  cost  price  does  he  receive? 

10.  A,  B and  C each  invest  $1500  in  business.  If  their  gain  during  a 
business  season  is  224%,  what  total  amount  of  gain  will  they  have  to  share? 

11.  A merchant  bought  a bill  of  goods  amounting  to  $1980,  for  which  he 
received  a commission  of  24%  ; what  was  the  amount  of  his  commission  ? 

12.  A bill  collector  having  bills  amounting  to  $897.62  to  collect,  gets  the 
money  for  78%  of  the  whole  amount,  and  receives  for  his  services  2%  of  the 
amount  collected.  How  much  does  he  receive  ? 

13.  In  a town  of  2748  inhabitants,  66f  % of  the  population  is  white  and 
the  remainder  colored.  How  many  colored  persons  are  there  in  the  town  ? 

lip.  A mine  produces  380  tons  of  metal ; 70%  of  the  metal  is  silver  and  30% 
lead.  Find  number  of  pounds  of  each. 


144 


PERCENTAGE 


15.  In  1875  the  population  of  a certain  town  was  1864.  In  1885  it  had 
gained  25%  over  1875;  and  in  1895  it  had  gained  20%  over  1885.  What  was 
the  population  in  1895  ? 

16.  In  1895  a certain  official’s  salary  was  $1800.  In  1896  his  salary  was 
raised  ll-g-%,  and  in  1897  it  was  raised  10%.  What  did  his  salary  average  for 
the  three  years? 

17.  A real  estate  agent  sells  a house  for  $9685  and  receives  2%  for  his  com- 
mission. What  does  the  owner  realize  on  the  sale  ? 

18.  A man,  having  received  a legacy  of  $42318,  invested  50%  of  it  in  stocks 
and  bonds,  paid  25%  of  the  remainder  for  a house  and  lot,  loaned  on  a mortgage 
33J%  of  what  he  then  had  left,  spent  20%  of  the  balance,  and  then  deposited  the 
rest  of  his  money  in  bank.  How  much  did  lie  deposit? 

19.  I sold  a consignment  of  goods  for  $1262.50;  wThat  was  my  commission 
at  1J%? 

20.  What  is  the  difference  between  16f  % of  $6248.40  and  62J%  of  $1678.50? 

21.  A grocer  bought  75  sacks  of  salt  at  $2.40  per  sack  and  65  barrels  of  flour 
at  25%  more  per  barrel  than  the  price  of  the  salt  per  sack  ; for  which  did  he  pay 
the  greater  amount,  and  how  much? 

22.  A man  owing  a debt  amounting  to  $186.20,  paid  20  % of  the  debt,  and 
afterward  35%  of  what  remained  unpaid  ; how  much  does  he  still  owe? 

23.  A certain  year  a farmer  raised  90  bushels  of  potatoes  to  the  acre  on  45 
acres  planted  ; the  next  year  the  yield  per  acre  was  10%  less,  and  the  number 
of  acres  planted  was  20%  more  ; how  many  bushels  did  he  raise  the  second  year? 

21^.  A farmer  planted  an  orchard  of  300  apple  trees,  25%  more  peach  trees 
than  apple  trees,  20%  fewer  plum  trees  than  peach  trees,  40%  fewer  cherry  trees 
than  apple  and  peach  trees  together;  how  many  trees  in  the  orchard? 

25.  A merchant  begins  business  with  a capital  of  $16000;  the  first  year  he 
gains  20%,  the  second  year  he  loses  25%,  the  third  year  he  gains  2%,  and  the 
fourth  year  he  gains  10%.  If  no  money  has  been  withdrawn,  what  is  his  capital 
at  the  end  of  the  fourth  year  ? 

26.  A’s  sales  for  a certain  year  are  20  % less  than  B’s ; B’s  sales  are  40  % more 
than  C’s;  C’s  sales  are  25%  less  than  D’s.  If  D’s  sales  are  $6000,  what  are  the 
sales  of  each  of  the  others  ? 

27.  How  much  linseed  oil  can  be  extracted  from  1270  pounds  of  flaxseed,  if 
flaxseed  contains  11%  of  oil  and  a pint  of  oil  weighs  f of  a pound? 

28.  18%  of  a man’s  wealth  is  in  real  estate,  24%  in  bank  stock.  26%  in  rail- 
road bonds,  and  the  remainder  in  money.  What  amount  is  invested  in  each,  and 
what  amount  has  he  in  money  if  his  wealth  amounted  to  $10288? 

29.  A,  B and  C are  partners.  A’s  investment  is  30%  more  than  B’s,  and 
C’s  investment  is  40  % less  than  A’s.  If  B’s  investment  is  $3500,  what  is  the  capital 
of  the  firm  ? 

30.  Two  railroad  companies  transported  5300  lbs.  of  freight  at  the  through 
rate  of  35  cents  per  cwt.  If  one  company  receives  45%  of  the  through  rate,  how 
much  should  each  receive? 


PERCENTAGE 


145 


515.  To  find  the  base,  the  percentage  and  rate  being  given. 

Formula. — Percentage  h-  Rate  (expressed  decimally)  = Base. 

Example. — A’s  farm  is  worth  $4050  which  is  35%  more  than  B’s  farm  is 
worth.  What  is  the  value  of  B’s  farm  ? 

T> 

Explanation. — If  A’s  farm  is  worth  35%  more  than  B’s,  then  100%  or 
B’s  + 35%  of  B’s  or  135%  of  B’s=$4050  or  A’s  farm.  If  135%  of  B’s  is  $4050,  1-35)  4050.00  P. 

100%,  or  B’s,  is  $3000.  30  00  B. 

516.  Rule. — To  find  the  base,  divide  the  percentage  by  the  rate  expressed  decimally. 

Note. — It  is  frequently  convenient  to  change  the  rate  to  its  equivalent  common  fraction  and 
solve  by  analysis. 


MENTAL  PROBLEMS 

517.  1.  A jeweler  sold  a watch  for  $30  and  thereby  gained  25  per  cent.;  what 
was  the  cost  of  the  watch? 

Solution. — The  gain  equals  1 of  the  cost,  which,  added  to  the  cost,  is  f of  the  cost,  or  $30  ; 1 
of  the  cost  equals  1 of  $30  or  $6,  and  the  whole  cost  equals  4 times  $6  or  $24. 

2.  A merchant  sold  a shawl  for  $14,  which  was  at  a gain  of  40  per  cent.; 
what  did  the  shawl  cost  him  ? 

3.  A farmer  sold  a cow  for  $32,  and  thereby  gained  33^  per  cent,  of  the 
cost;  what  was  the  cost  of  the  cow? 

It..  An  agent  sold  a library  for  $120,  thereby  losing  33^  per  cent,  of  its  value  ; 
what  was  its  value? 

5.  A merchant  sold  cloth  at  $6  per  }rard,  thereby  gaining  20  per  cent.;  how 
should  he  have  sold  it  to  gain  40  per  cent.? 

6.  If  by  selling  land  at  $75  per  acre,  25  per  cent,  was  gained,  what  was  its 
cost  ? 

7.  On  muslin  sold  at  9 cents  a yard,  50  per  cent,  was  gained  ; how  much  did 
it  cost  a yard  ? 

8.  A boat  was  sold  for  $70,  which  was  at  a loss  of  16f  per  cent.;  at  what  price 
should  it  have  been  sold  to  gain  16f  per  cent.? 

9.  A wagon  was  sold  for  $90,  which  was  10  per  cent,  below  its  value  ; what 
would  have  been  gained  by  selling  it  for  $125? 

10.  A dealer  sold  2 wheels  for  $30  each.  On  the  one  he  gained  25  per  cent, 
and  on  the  other  he  lost  25  per  cent. ; how  much  did  he  gain  or  lose  by  the 
transaction  ? 

11.  A man  gained  25  per  cent,  by  selling  a watch  for  $25  more  than  it  cost ; 
required  the  cost. 

12.  A hat  was  sold  for  25  cents  more  than  its  cost,  which  was  a gain  of  10  per 
cent.;  what  was  its  cost  ? 

13.  A watch  was  sold  at  a loss  of  33^  per  cent.  If  the  price  received  was  $60, 
what  was  its  cost? 


146 


PERCENTAGE 


14-  A man  gained  $20  by  selling  a boat  for  25  per  cent,  more  than  it  cost; 
what  would  he  have  gained  by  selling  it  for  $90? 

15.  A piano  was  sold  for  $50  less  than  its  value,  which  was  a loss  of  12|  per 
cent. ; what  would  have  been  the  gain  if  it  had  been  sold  for  $440  ? 

16.  A certain  kind  of  merchandise  yields  5 per  cent,  profit ; how  much  should 
I buy  to  gain  $90  ? 

17.  I received  $300  or  16§  per  cent,  of  an  amount  owed  to  me  ; what  was  the 
amount  of  the  debt? 

18.  A merchant  marks  goods  at  an  advance  of  20  per  cent.;  what  is  the  cost 
of  goods  marked  $360  ? 

19.  Carpets  sold  at  $125  make  a gain  of  66§  per  cent;  what  was  the  cost  of 
the  carpets  ? 

20.  A man  offers  to  pay  $500  for  a 25  per  cent,  interest  in  a business ; what  is 
the  valuation  of  the  business  ? 

WRITTEN  EXERCISE 

518.  1 ■ $425  is  5 % of  what? 

2.  720  lbs.  is  9 % of  what  ? 

3.  740  is  7J%  less  than  what  number  ? 

4.  $817.32  is  6J%  of  what  amount  ? 

5.  What  number  increased  by  12J%  of  itself  equals  896? 

6.  $1252.16  is  8%  more  than  what  amount? 

7.  $32.87  is  %%  of  what  sum  ? 

8.  What  sum  diminished  by  3%  of  itself  equals  $1763.24? 

9.  $9378.25  is  87f  % of  how  much  ? 

10.  12  yds.  is  \ % of  what  distance  ? 

11.  5 bu.  2 pk.  is  11%  of  what  quantity? 

12.  12J%  of  an  amount  of  money  is  $250;  what  is  the  amount? 

13.  If  12J%  of  the  distance  from  Philadelphia  to  New  York  is  11-J  miles, 
what  is  the  whole  distance? 

14-  19  bus.  are  66f%  of  what  number  of  bushels? 

15.  $10  is  2J%  more  than  what  sum  ? 

16.  666f  is  66f%  less  than  what  number? 

17.  The  percentage  is  11  and  the  rate  is  18f  % ; what  is  the  base? 

18.  990  miles  is  16f%  of  how  many  miles? 

19.  £20  4s.  is  30%  of  how  many  pounds  sterling? 

20.  $36  is  10%  more  than  15%  of  what  number? 

21.  4 mi.  18  rd.  12  ft.  is  18f  % of  how  many  miles? 

22.  lS-g-%  of  a man’s  salary  is  $300;  what  is  his  salary? 

23.  What  number  will  leave  740  after  deducting  7J%  of  it? 

24 • Through  a reduction  of  28f%  from  list  price,  I gain  $40  on  a purchase. 
W1  lat  is  the  list  price? 


PERCENTAGE 


147 


WRITTEN  PROBLEMS 

519.  1 ■ If  24%  of  a contract  can  be  executed  in  108  days,  in  what  time  can 
the  whole  contract  be  executed  ? 

2.  A man  owing  a bill  pays  35%  of  it  by  paying  $83.65.  How  much  does 
he  still  owe? 

3.  A man  received  a cash  discount  of  $60  on  a bill  at  the  rate  of  2%  ; 
what  was  the  amount  of  the  bill? 

J.  A man  bought  a table  at  a discount  of  124%  from  the  marked  price  and 
saved  $4;  what  was  the  price  of  the  table? 

5.  A man  obtained  $660  for  goods  sold,  which  was  at  a gain  of  20%  to 
him  ; what  had  he  paid  for  the  goods? 

6.  A farm  was  sold  for  $18000,  which  was  15%  less  than  the  price  asked 
for  it ; what  was  asked  for  it  ? 

7.  Jones  and  Smith  together  own  $24500,  and  Smith  owns  16§  % less  than 
Jones;  how  much  does  each  own? 

8.  If  I sell  $30000  worth  of  property,  what  is  left  will  be  worth  85%  of  the 
value  of  the  whole  property ; what  is  the  value  of  the  property  I have  left? 

9.  $1980  is  20%  of  the  cost  of  B’s  farm ; and  the  cost  of  B’s  farm  is  90%  of 
the  cost  of  A’s  farm.  How  much  more  did  A’s  farm  cost  than  B’s? 

10.  Brown  has  his  property  insured  for  80%  of  its  value.  If  his  property 
is  damaged  25%  by  fire,  and  the  insurance  company  pays  him  $2340,  being  25% 
of  his  policy,  how  much  does  he  lose  by  the  fire? 

11.  A broker’s  commission,  at  |%,  amounted  to  $62.50;  what  was  the 
amount  of  the  transaction? 

12.  By  cutting  down  his  expenses  3^%,  a merchant  saves  $482.18.  What 
are  his  expenses? 

13.  A man  owning  f of  a ship,  sells  274%  of  his  share  for  $680.  What  is  the 
value  of  the  whole  vessel  at  that  rate? 

7J.  A sold  his  horse  for  $7840,  which  was  164%  more  than  it  cost.  What 
did  it  cost? 

15.  174%  °f  Jones’s  stock  of  goods  was  destroyed  by  fire,  and  he  received 
$2162.18  insurance,  which  was  f of  the  loss.  What  was  his  whole  stock  worth  ? 

16.  After  deducting  3%  for  prompt  payment,  a merchant  receives  $1284.92 
for  a bill  of  goods.  What  was  the  whole  amount  of  the  bill  ? 

17.  A’s  sales  for  the  year  are  13%  less  than  B’s,  and  B’s  sales  are  llf% 
more  than  C’s.  A’s  sales  amount  to  $6289.17.  Find  B’s  and  C’s  sales. 

18.  Smith’s  share  of  a certain  claim  was  35%.  After  paying  5 % for  having 
the  claim  collected,  Smith  received  $2182.93.  What  was  the  amount  of  the  entire 
claim  ? 

19.  In  a certain  orchard  there  are  25%  more  cherry  trees  than  peach  trees, 
20%  fewer  plum  trees  than  cherry  trees,  25%  more  pear  trees  than  plum  trees, 
and  20%  more  apple  trees  than  pear  trees.  There  are  60  apple  trees.  How  many 
trees  are  there  in  all  ? 


148 


PERCENTAGE 


30.  A bankrupt  pays  liis  creditors  51yf  cents  on  the  dollar.  Barnes  & Co. 
receive  $822.18 ; what  was  the  amount  of  their  claim  ? 

SI.  A man  marked  his  goods  25%  more  than  cost;  he  sold  them  for  $376, 
which  was  6%  less  than  the  marked  price.  Find  his  gain  or  loss. 

SS.  A dealer  marked  his  goods  30%  above  cost.  The  goods  becoming 
damaged,  he  sold  them  for  20%  less  than  the  marked  price;  his  customer  failed 
and  he  lost  20%  of  his  selling  price.  If  he  received  $800,  find  his  net  loss. 

S3.  A bought  a house,  paying  $1610  cash  and  giving  his  note  for  the 
remainder,  which  was  12J%  of  the  cost.  What  did  the  house  cost,  and  what  was 
the  amount  of  the  note? 

SI/..  In  building  a house,  45%  of  the  cost  was  for  brickwork,  25%  for  the 
carpenter  work,  15%  for  the  mason  work,  and  the  remainder,  amounting  to 
$3840,  was  for  painting  and  plastering;  what  did  the  bouse  cost,  and  what  was 
the  amount  paid  for  each  part  of  the  work? 

35.  Of  a cargo,  35%  was  sold  to  one  buyer,  60%  of  the  remainder  to  another, 
and  the  remainder,  amounting  to  1716  tons,  to  a third.  How  many  tons  in  the 
cargo  ? 

36.  A,  B and  C formed  a partnership.  A invested  $25200,  which  is  20% 
more  than  B invested,  and  20%  less  than  C invested.  Find  the  capital  of  the 
firm. 

37.  A owns  15%  of  a business;  B,  25%;  C,  28%;  and  D,  the  remainder. 
What  is  the  value  of  each  one’s  share,  if  D’s  share  is  $34464? 

38.  A paid  $3402  for  a house  which  was  20%  less  than  it  cost  to  build  it ; 
what  did  it  cost  to  build  the  house? 

39.  Mr.  Jones  withdrew  30%  of  his  deposit  from  bank,  and  invested  30% 
of  what  he  withdrew,  in  a house  which  cost  $3000.  What  amount  of  money  has 
he  remaining  in  bank  ? 

30.  A bought  75  bbls.  of  flour,  and  B increased  his  stock  22%  by  buying 
12%  fewer  than  A.  How  many  barrels  has  B now  ? 

520.  To  find  the  rate,  the  percentage  and  base  being  given. 

Formula. — Percentage-!- Base=Rate  (expressed  decimally). 

Example. — Bread  made  from  120  pounds  of  flour  weighs  162  pounds.  What 
per  cent,  more  than  the  flour  does  the  bread  weigh? 

Explanation. — More  than  follows  the  words  per  cent,  and  precedes 
the  base  (the.  flour).  Hence,  the  flour,  120  pounds,  is  the  base.  Subtracting 
the  weight  of  the  flour  from  the  weight  of  the  bread  to  find  how  much  more 
the  bread  weighs  than  the  flour,  gives  42  pounds  as  the  percentage.  Dividing 
the  percentage  42  pounds  by  the  base  120  pounds,  carrying  the  quotient  to 
hundredths,  will  give  .35  or  35%. 

521.  Rule. — To  find  the  rate  divide  the  percentage  by  the  base.  Or,  find 
equivalent  fraction  and  change  to  a decimal. 


162 

120 

1.20)42.00 

.35  or  35  % 


PERCENTAGE 


149 


MENTAL,  PROBLEMS 

522.  1.  A man  bought  a bicycle  for  $50  and  sold  it  for  $60 ; what  was  his 
gain  per  cent.  ? 

Solution. — He  gained  $10,  which  is  \ of  the  cost ; this  is  the  equivalent  fraction  for  20  per  cent. 

2.  A merchant  sold  knives  at  30  cents  each  which  cost  him  25  cents  each  ; 
what  was  his  gain  per  cent.? 

3.  A merchant  sold  shawls  for  $8  which  cost  him  $6 ; what  was  his  gain 
per  cent.  ? 

If.  A boatman  paid  $20  for  a boat  and  sold  it  for  $15 ; what  was  his  loss 
per  cent.? 

5.  A farmer  bought  a horse  for  $120  and  sold  it  for  f-  of  the  cost;  required 
the  loss  per  cent. 

6.  A dairyman  bought  10  cows  for  $240,  and  sold  8 of  them  for  what  all  cost ; 
what  was  the  gain  per  cent.? 

7.  A boy  having  50  cents  spent  20  per  cent,  of  it ; what  per  cent,  remained  ? 

8.  I own  property  worth  $5000,  the  yearly  repairs  of  which  cost  me  $150  ; 
what  per  cent,  of  the  value  is  expended  in  repairs? 

9.  To  close  out  a stock  of  goods  a merchant  offers  $500  worth  for  $400 ; 
what  is  the  per  cent,  of  discount  ? 

10.  The  marked  price  on  chairs  is  $2  each,  and  they  are  sold  at  $1.75  ; what 
is  the  per  cent,  of  discount  ? 

11.  On  a debt  of  $750  a man  paid  $375 ; what  per  cent,  of  his  debt  did 
he  pay  ? 

12.  A merchant  buys  a bill  of  goods  for  $400,  and  receives  a discount  of 
$20  for  cash  ; what  is  the  rate  of  cash  discount? 

13.  An  agent  sells  a consignment  of  goods  for  $1200;  if  his  commission  is 
$30,  what  is  the  rate  of  commission  ? 

Ilf.  A stockholder  of  a bank,  who  owns  $1200  worth  of  stock,  is  assessed 
$240  ; what  is  the  rate  of  assessment? 

15.  A stockholder  of  a railroad  company,  owning  $12000  worth  of  stock, 
received  a dividend  of  $360  ; what  is  the  rate  of  dividend  ? 

16.  In  a roll  of  carpet  of  40  yards,  4 yards  are  wasted  in  matching  ; what  is 
the  per  cent,  of  waste? 

17.  Linoleum  costing  60  cents  a yard  is  sold  for  54  cents  a yard  ; what  is  the 
loss  per  cent.  ? 

18.  Overcoats  marked  at  $25  have  been  reduced  to  $15  ; what  is  the  per  cent, 
of  reduction  ? 

19.  A merchant  who  has  a stock  of  $6000  gains  $2000  per  year  upon  his 
stock  ; what  is  his  average  rate  of  gain  ? 

20.  A house  valued  at  $4000,  rents  for  $500  a year ; what  per  cent,  of  its 
value  is  received  in  rent? 


150 


PERCENTAGE 


WRITTEN  EXERCISE 

523.  1.  $309.68  is  what  % of  $13272? 

2.  49  lb.  8 oz.  is  what  % of  450  lb.? 

3.  $823  is  what  % more  than  $712  ? 
lh  What  % less  than  1262  is  928? 

5.  554f  lbs.  is  what  % of  3 long  tons  ? 

6.  10  hr.  57  min.  is  what  % of  a year? 

7.  $854.44  is  what  °/c  of  $6252? 

8.  $2906.86  is  what  % less  than  $3620? 

9.  $32899.40  is  what  °/0  more  than  $24390  ? 

10.  19  yd.  2 ft.  44  in.  is  what  % of  a mile? 

11.  What  % of  $120  is  $18? 

12.  What  % of  2000  pounds  is  600  pounds? 

13.  What  % of  200  acres  is  10  acres,  20  sq.  rods  ? 

Ilf..  What  °/o  of  a day  is  5 hr.  30  min.  ? 

15.  What  % of  a short  ton  is  196  pounds? 

16.  What  °/0  of  3.1416  is  .7854? 

17.  What  % of  3.1416  is  .5236? 

18.  What  °/0  of  .7854  is  .5236? 

19.  What  % of  2156.42  is  231? 

20  What  % of  $3000  is  $150? 

WRITTEN  PROBLEMS 

524.  1.  Sold  goods  for  $2770  that  cost  $2400;  what  was  the  rate  per  cent, 
of  profit ? 

2.  What  % of  £12  6s.  4 d.  is  £2  3s.  2d.? 

3.  A loss  of  2s.  on  a pound  sterling  is  what  rate  per  cent,  of  loss? 

If.  Goods  costing  $840  were  sold  for  $720;  at  what  rate  above  the  selling 
price  should  they  have  been  sold  to  gain  $120? 

5.  Sold  goods  for  $3200  that  cost  $2600 ; what  was  the  per  cent,  of  gain  ? 

6.  What  per  cent,  of  a short  ton  is  a long  ton  ? 

7.  What  per  cent,  of  a meter  is  a yard? 

8.  What  per  cent,  of  a mile  is  a rod  ? 

9.  What  per  cent,  of  a dollar  is  a franc? 

10.  What  per  cent,  of  a pound  sterling  is  a dollar? 

11.  What  per  cent,  commission  does  a collector  charge,  if  he  remits  me 
$277.68  after  collecting  a bill  of  $284.80? 

12.  Sold  goods  for  $878.22  that  cost  $714.  What  was  my  gain  per  cent.? 

13.  A baseball  team  lost  12  games  out  of  32.  What  per  cent,  did  they  win  ? 
Ilf  Jones  pays  $152.32  for  $4760  insurance.  What  rate  does  he  pay? 


PERCENTAGE 


151 


15.  The  population  of  a certain  town  was  2764  in  1880.  In  1890  it  was 
3173.  What  was  the  per  cent,  of  increase? 

16.  Imported  1230  yards  of  dress  goods  at  $1.32,  and  paid  $649.44  duty. 
What  per  cent,  was  the  duty  ? 

17.  Of  a lot  of  435  crates  of  fruit,  60  crates  spoiled.  What  per  cent,  must 
the  price  of  the  remainder  be  advanced  to  cover  the  loss  ? 

18.  If  12340  lbs.  of  an  alloy  containing  4%  nickel  is  mixed  with  4560 
lbs.  of  another  alloy  containing  7f%  nickel,  what  per  cent,  of  nickel  does  the 
mixture  contain  ? 

19.  If  A receives  a commission  of  for  selling  goods,  and  B receives 
$54.16§  more  than  A on  each  $5000  worth  sold,  what  is  the  rate  of  B’s 
commission  ? 

50.  What  per  cent,  more  than  $3875.60  is  93J%  of  $8972.10? 

51.  A house  costs  $4000,  and  rents  for  $25  per  month ; the  expenses  for  the 
year  are,  water  tax,  $15 ; city  taxes,  $80 ; and  repairs,  $25.  What  net  yearly 
rate  per  cent,  on  the  investment  does  the  house  pay  ? 

SS.  A has  $6500,  B has  28%  more  money  than  A.  What  per  cent,  less  than 
B has  A ? 

S3.  A father  receives  a yearly  salary  of  $1250  and  his  son  a yearly  salary  of 
$1000.  What  per  cent,  does  the  father  get  more  than  the  son  ? 

Slf..  The  list  price  of  a piano  is  $550.  It  can  be  bought  for  $539.  What  per 
cent,  is  deducted  from  the  list  price  ? 

55.  A man’s  salary  is  $2750  per  year;  he  pays  $343.75  for  rent  and  $412.50 
for  other  expenses.  What  per  cent,  of  his  salary  does  he  save? 

56.  A jeweler  bought  a watch  for  $160  and  asked  such  a price  for  it  that 
after  falling  $18,  he  still  made  20  per  cent.  What  per  cent,  did  he  deduct  from 
his  asking  price  ? 

57.  A house  which  costs  $9000  rents  for  $50  per  month.  What  is  the  net 
annual  rate  per  cent,  on  the  investment  if  the  expenses  are,  $130  for  taxes,  $25 
for  insurance,  $175  for  ground  rent  and  $45  for  repairs  ? 

58.  Bought  450  bushels  of  wheat  at  $1.25  a bushel.  If  10  bushels  spoiled, 
and  I sold  the  remainder  at  $1.40  a bushel,  what  was  my  gain  per  cent.? 

59.  If  I sell  f of  an  article  for  what  f-  of  it  cost,  what  is  my  gain  or  loss  per 
cent.  ? 

30.  If  an  article  bought  at  20%  below  asking  price  is  sold  for  20%  more 
than  asking  price,  what  per  cent,  is  gained? 


PROFIT  AND  LOSS 

525.  Calculations  in  Profit  and  Loss  are  computations  to  determine  the 
gain  or  loss  resulting  from  buying  merchandise  at  one  price  and  selling  at 
another. 

526.  The  cost  of  goods  is  the  amount  paid  to  purchase  them,  or  to  produce 
them. 

527.  The  prime  cost  of  goods  is  the  first  or  original  cost. 

528.  The  gross  cost  is  the  prime  cost  increased  by  all  incidental  expenses, 
such  as  freight,  commission,  insurance,  duty,  packing,  drayage,  etc. 

529.  The  selling  price  of  goods  is  the  price  at  which  they  are  sold. 

530.  The  net  selling  price  is  the  original  selling  price  diminished  by  any 
deductions  or  allowances  made  to  the  customer. 

531.  Profit  is  the  difference  between  cost  and  selling  price,  the  selling  price 
being  more  than  the  cost. 

532.  Loss  is  the  difference  between  cost  and  selling  price,  the  selling  price 
being  less  than  the  cost. 

533.  Gain  or  loss  in  commercial  transactions  is  reckoned  as  a certain  per 
cent,  of  the  gross  cost. 

534.  To  find  the  profit  or  loss  when  the  cost  and  rate  are  given. 

Example. — For  how  much  must  goods  that  cost  §34  a dozen  be  sold  to 
realize  a profit  of  15%  ? 

$34 
.15 

170 

34 

$5.10  profit 

535.  Rule. — Multiply  the  cost  by  the  rate  expressed  decimally. 

MENTAL,  PROBLEMS 

536.  1.  A desk  costing  $60  was  sold  at  33^  per  cent,  profit;  what  was  the 
amount  of  gain  ? 

Solution. — A gain  of  33}  per  cent,  is  a gain  of  1 of  the  cost,  or  $20. 

2.  A watch  costing  $20  was  sold  at  a gain  of  20  per  cent. ; what  was  the 
selling  price  ? 

3.  A horse  costing  $80  was  sold  at  a loss  of  25  per  cent.;  what  was  the 
amount  of  loss  ? 

4-  Land  costing  $70  an  acre  increases  in  value  14f  per  cent.  ; what  is  it 
worth  an  acre  ? 

5.  A clock  costing  $36  is  sold  at  a loss  of  16f  percent.;  what  is  the  amount 
of  loss  ? 


$34  cost 
5.10  profit 

$39.10  selling  price 


152 


PROFIT  AND  LOSS 


153 


6.  Bank  stock  whose  par  value  is  $50  is  quoted  at  60  per  cent. ; what  is  a 
share  worth  ? 

7.  Carpet  marked  85  cents  a yard  is  sold  at  a discount  of  20  per  cent. ; 
what  is  the  selling  price  per  yard? 

8.  A man  who  pays  $16  a month  rent  finds  it  increased  12J  percent. ; what 
must  he  now  pay  per  month  ? 

9.  A rug  costing  $32  is  marked  37J  per  cent,  above  cost ; what  is  its  selling 
price  ? 

10.  Stock  costing  me  $35  a share  increases  in  value  10  percent. ; how  much 
shall  I gain  per  share  by  selling? 

11.  Bought  a horse  for  $100  and  asked  for  him  20  per  cent,  above  cost ; 
what  would  I gain  by  accepting  10  per  cent,  less  than  I ask  ? 

12.  I buy  flour  at  $5  a barrel  and  sell  for  20  per  cent,  more ; what  do  I get 
per  barrel  ? 

13.  Silk  bought  at  $2  a yard  is  marked  50  per  cent,  above  cost  and  sold  33  J 
per  cent,  below  marked  price  ; what  is  gained  on  a yard  ? 

ll^.  Lamps  costing  $5  each  are  marked  60  per  cent,  above  cost  and  sold  at  a 
discount  of  25  per  cent,  off  marked  price;  what  is  gained  on  a lamp? 

15.  Cloth  marked  $6  a yard  is  bought  at  a discount  of  16§  per  cent,  and  sold 
at  a gain  of  20  per  cent. ; what  is  the  gain  on  a yard  ? 

16.  Tea  costing  45  cents  a pound  is  sold  20  per  cent,  above  cost ; what  is  the 
selling  price  of  a pound  ? 

17.  Wheat  quoted  at  80  cents  a bushel  drops  5 per  cent. ; if  I buy,  and  sell 
again  when  it  reaches  80  cents,  how  much  do  I make  on  a bushel? 

18.  Bought  coffee  worth  32  cents  a pound  at  a discount  of  12J  per  cent.  ; 
what  is  saved  on  a pound  ? 

19.  A piano  listed  at  $300  is  sold  at  33J  per  cent,  below  list ; what  is  received 
for  it  ? 

20.  A piano  costing  $200  to  build  is  listed  at  $400  and  sold  40  per  cent. 
below1  list;  what  is  the  amount  of  gain  ? 

ORAL  EXERCISE 

537.  1.  For  how  much  must  goods  be  sold  that  cost  $120  to  gain  25%  ? 


2. 

Cost 

$220 

to 

gain 

50%  ? 

12. 

Cost 

$90  to  lose  2J%  ? 

3. 

Cost 

$375 

to 

gain 

33J%? 

13. 

Cost 

$90  to  gain  13^%  ? 

4- 

Cost 

$144 

to 

gain 

16f  % ? 

n. 

Cost 

$120  to  lose  12J  % ? 

5. 

Cost 

$250 

to 

gain 

12%  ? 

15. 

Cost 

$144  to  gain  6f%  ? 

6. 

Cost 

$117 

to 

gain 

5%  ? 

16. 

Cost 

$30  to  lose  2J  % ? 

7. 

Cost 

$345 

to 

gain 

20  %? 

17. 

Cost 

$70  to  gain  14f  % ? 

8. 

Cost 

$110 

to 

gain 

15%  ? 

18. 

Cost 

$90  to  lose  ll-g-%  ? 

9. 

Cost 

$400 

to 

lose  '• 

30%  ? 

19. 

Cost 

$300  to  lose  33J  % ? 

10. 

Cost 

$1100  to  lose 

! 2%  ? 

20. 

Cost 

$200  to  gain  40  % ? 

11. 

Cost 

$120 

to 

gain 

8i%? 

154 


PROFIT  AND  LOSS 


WRITTEN  PROBLEMS 

538.  1 . Fish  bought  for  $640  were  sold  at  an  average  gain  of  24%  ; what 
was  received  for  them  ? 

2.  A fur  dealer  marked  down  his  stock  15%  ; what  would  he  the  reduced 
price  of  a set  formerly  marked  $117.50? 

3.  The  list  price  of  a lot  of  goods  is  $960 ; if  I buy  at  10%  off  and  sell  at 
a profit  of  30%,  what  do  I gain  on  the  goods? 

1^.  A vessel  cost  its  owner  to  build  $21000,  and  was  sold  by  him  at  a loss  of 
15%  ; the  buyer  sold  it  again  at  15%  above  what  he  paid  for  it;  how  did  the 
price  he  received  for  it  compare  with  its  original  cost? 

5.  A merchant  began  business  with  a capital  of  $17500  and  gained  16|%> 
which  he  added  to  his  capital ; the  second  year  he  lost  2%  ; what  is  his  capital 
at  the  end  of  the  second  year? 

6.  A man  doing  a business  of  $6500  a year  finds  his  expenses  are  35%  of 
his  business;  what  is  his  net  business? 

7.  A man’s  inventory  of  furniture  and  fixtures  at  the  beginning  of  his 
business  year  is  $1550;  at  the  end  of  the  year  he  marks  off  16|  % for  wear  and 
tear ; what  does  the  new  inventory  show  ? 

8.  A man  buys  merchandise  to  the  amount  of  $72500  and  averages  a profit 
of  15%  ; what  is  the  amount  of  sales? 

9.  A company  having  a capital  of  $250000  has  gained  4%  on  its  capital 
and  divides  2J%  of  its  capital  in  dividends;  how  much  of  the  profit  is  undi- 
vided ? 

10.  Goods  that  cost  $375.84  were  marked  30%  more  than  cost  and  were  sold 
for  15%  less  than  the  marked  price.  What  was  the  net  gain? 

11.  What  must  be  the  selling  price  of  goods  that  cost  $3642.70,  to  net  a 
profit  of  18%  ? 

12.  A merchant  bought  goods  for  $4286.91  and  sold  them  at  a loss  of  224%. 
How  much  did  he  receive  for  the  goods  ? 

13.  Sold  goods  that  cost  $237.28  at  a profit  of  25%,  but  my  customer  failed 
and  paid  only  75  cents  on  the  dollar;  did  I gain  or  lose  on  the  goods,  and  how 
much  ? 

Ilf.  Bought  three  lots  of  goods  for  $2780.14,  $3628.32  and  $4245.79,  respect- 
ively. The  first  I sold  at  a profit  of  12J%,  the  second  at  a loss  of  374%  and  the 
third  at  a profit  of  30%.  What  was  my  gain  or  loss  on  the  whole  ? 

15.  A hardware  dealer  bought  6 gross  of  locks  at  $4.80  a dozen  and  marked 
them  40%  above  cost.  If  he  sells  them  at  12%  less  than  the  marked  price,  what 
will  be  his  profit  on  the  lot  ? 

16.  A bought  a house  and  lot  for  $6500  and  sold  it  to  B at  a loss  of  124%  ; 
B sold  it  for  22J%  more  than  he  paid  for  it;  how  much  did  he  receive? 

17.  A dry  goods  merchant  bought  348  yards  of  cloth  for  $131.37  and  sold 
4 of  it  at  a profit  of  11J  cents  a yard  ; he  then  sold  the  remainder  at  such  a price 
as  to  gain  35  % on  the  whole.  At  what  price  per  yard  did  he  sell  the  remainder? 


PROFIT  AND  LOSS 


155 


18.  A bought  goods  for  $2138.75  and  sold  them  to  B at  a profit  of  18f  % ; 
B sold  them  to  C at  a profit  of  14-f-%  ; C sold  them  to  D at  a loss  of  20J%  and 
D sold  them  at  a profit  of  8%.  How  much  did  D receive  for  them? 

19.  A cargo  of  wheat,  weighing  632897  lbs.  was  bought  for  68f  cents  a 
bushel.  One-third  of  it  was  sold  at  a profit  of  8%,  three-eighths  of  the 
remainder  at  a profit  of  12£%  and  the  balance  at  a loss  of  3J%.  The  proceeds 
were  invested  in  coffee,  which  was  sold  at  a profit  of  18f-%.  What  was  the 
entire  gain  ? 

SO.  Jones  bought  a house  for  $3285  and  sold  it  to  Smith  at  a profit  of  94%. 
Smith  spent  $394.28  for  repairs  and  then  sold  the  house  at  a profit  of  15%.  How 
much  did  he  gain  ? 

539.  To  find  the  rate  per  cent,  when  the  profit  or  loss  and  the  cost 
are  given. 


Example. — What  is  the  rate  per  cent,  of  profit  on  goods  bought  for  $6520 
and  sold  for  $7458.88  ? 


$7458.88  selling  price 
6520.  cost 


14f  or  14f  % 
6520)938188 
6520 


$ 938.88  profit 


28688 

26080 


1304 


j 26.08 
6520 


2 

5 


540.  Rule. — Divide  the  profit  or  loss  by  the  cost. 


MENTAL  PROBLEMS 

541.  1.  Damaged  merchandise  costing  $240  is  sold  for  $180;  what  is  the 
per  cent,  of  loss  ? 

Solution. — There  is  a loss  of  $60,  or  of  or  1 of  the  value,  which  is  equivalent  to  25  per 
cent.  loss. 

S.  What  per  cent,  is  gained  by  selling  an  article  costing  50  cents  for  60 
cents  ? 

3.  What  per  cent,  is  lost  by  selling  an  article  costing  60  cents  for  50  cents? 

J.  If  wheat  is  bought  for  80  cents  and  sold  the  same  day  for  84  cents,  what 
is  the  rate  of  gain  ? 

5.  If  wheat  is  bought  at  80  cents  and  sold  the  same  day  for  76  cents,  what 
is  the  rate  of  loss  ? 

6.  In  an  invoice  of  eggs  two  out  of  a dozen  are  broken;  what  is  the  per 
cent,  of  breakage  ? 

7.  In  a box  of  oranges  one  out  of  a dozen  is  unsalable ; what  is  the  per 
cent,  of  loss  from  this  source  ? 

8.  In  sponging  cloth  there  is  a shrinkage  of  3 inches  to  the  yard;  what  is 
the  per  cent,  of  shrinkage  ? 


156 


PROFIT  AND  LOSS 


9.  If  an  article  costing  10  cents  is  marked  50  per  cent,  above  cost  and  sold 
for  3 cents  less  than  the  marked  price,  what  is  the  per  cent,  of  gain  on  the  sale  ? 

10.  A horse  costing  $200  is  sold  for  $250 ; what  is  the  gain  per  cent.? 

11.  A horse  costing  $200  is  sold  for  $150  ; what  is  the  loss  per  cent.? 

18-  What  per  cent,  is  gained  by  selling  an  article  costing  70  cents  for  77 
cents  ? 

13.  If  one-half  of  an  invoice  be  sold  at  25  per  cent,  gain  and  the  other  half 
at  20  per  cent,  loss,  what  is  the  rate  per  cent,  of  loss  or  gain  on  the  whole  invoice? 

H-  What  is  gained  or  lost  per  cent,  by  asking  30  per  cent,  more  than  cost 
for  goods  and  selling  them  for  10  per  cent,  less  than  the  asking  price  ? 

15.  What  do  I gain  per  cent,  by  selling  chairs  at  $3  each  which  cost  $2.50 
each  ? 

16.  If  an  article  be  marked  to  gain  20  per  cent,  and  ajdiscount  of  10  per  cent, 
be  allowed,  what  will  be  the  rate  per  cent,  of  gain  ? 

17.  If  a share  of  stock  costs  $50  and  the  market  price  is  $40,  what  has  been 
the  rate  per  cent,  of  shrinkage  in  its  value? 

18.  If  a share  of  stock  whose  par  was  100  is  bought  at  70  and  sold  at  80, 
what  is  the  gain  per  cent.? 

19.  What  do  I gain  per  cent,  by  selling  stock  at  60  which  I bought  at  56  ? 

20.  What  do  I earn  per  cent,  as  a commission  agent,  by  selling  $120  worth 
of  goods,  if  I receive  $6  commission  ? 

ORAL  EXERCISE 


542.  Tell  the  rate  per  cent,  of  profit  or  loss  on  the  following  : 


1. 

Cost 

$150,  sold 

for 

$200. 

11. 

Cost 

$500,  sold 

for  $625. 

2. 

Cost 

$66-§,  sold 

for 

$133|. 

12. 

Cost 

$600,  sold 

for  $720. 

3. 

Cost 

$375, sold 

for 

$625. 

13. 

Cost 

$100,  sold 

for  $90. 

4- 

Cost 

$600,  sold 

for 

$450. 

n. 

Cost 

$200,  sold 

for  $175. 

5. 

Cost 

$875,  sold 

for 

$750. 

15. 

Cost 

$300,  sold 

for  $250. 

6. 

Cost 

$144, sold 

for 

$216. 

16. 

Cost 

$600,  sold 

for  $642. 

7. 

Cost 

$225, sold 

for 

$150. 

17. 

Cost 

$850,  sold 

for  $765. 

8. 

Cost 

$80,  sold  for  | 

588.80. 

18. 

Cost 

$680,  sold 

for  $510. 

9. 

Cost 

$510,  sold 

for 

$680. 

19. 

Cost 

$216, sold 

for  $270. 

10. 

Cost 

$1200,  sold  for  $1176. 

20. 

Cost 

$350, sold 

for  $525. 

WRITTEN  PROBLEMS 

543.  1.  Merchandise  bought  for  $6250  was  sold  $782.50  above  cost;  wha 
was  the  per  cent,  of  gain  ? 

2.  Merchandise  bought  for  $6250  was  sold  for  $5467.50  ; what  was  the  per 
cent,  of  loss  ? 

3.  A business  takes  in  $45200  a year ; its  expenses  are  $6780 ; what  is  the 
rate  per  cent,  of  expense  ? 


PROFIT  AND  LOSS 


157 


4.  From  a tank  of  oil  containing  2250  gallons  there  is  a leakage  of  45 
gallons  a week  ; what  per  cent,  is  the  leakage  ? 

5.  A creditor  gave  a collector  $50  to  collect  $1000;  what  was  the  rate  of 
commission  ? 

6.  Carpets  bought  at  85  cents  a yard  are  marked  down  to  68  cents  ; what 
is  the  per  cent,  of  loss  ? 

7.  What  is  gained  per  cent,  by  marking  at  33J%  above  list  and  selling  at 
25%  off  the  marked  price? 

8.  What  is  the  gain  per  cent,  in  buying  at  10%  off  list  price  and  selling  at 
20%  above  list  price? 

9.  A house  was  offered  for  sale  at  25%  of  its  cost,  but  finding  no  takers  at 
that  price,  was  sold  at  25%  off  the  price  first  asked  ; what  was  the  rate  per  cent, 
of  gain  or  loss  ? 

10.  A man  paid  $650  for  a diamond  and  sold  it  for  $910;  what  did  he  gain 
per  cent.  ? 

11.  A lot  of  goods  bought  for  $3786  was  marked  25%  above  cost,  but 
afterwards  sold  for  25%  less  than  the  marked  price.  What  per  cent,  was  gained 
or  lost  ? 

12.  If  an  investment  of  $7463.80  yields  annually  $559.78,  what  is  the  rate 
per  cent,  of  income  ? 

13.  Bought  goods  for  $4987.36,  and  sold  them  at  an  advance  of  28f  %,  but 
was  unable  to  collect  $374.50  of  the  sales.  What  per  cent,  did  I gain  ? 

14-  AVhat  per  cent,  is  gained  by  buving  cigars  at  $65  a thousand  and  selling 
them  at  three  for  a quarter  ? 

15.  Merchandise  costing  $6387.50  was  sold  as  follows:  3-  at  a profit  of  42%, 

$1846.25  worth  (cost  price)  at  a profit  of  27%,  and  the  remainder  at  cost.  What 
per  cent,  was  gained  ? 

16.  A grocer  bought  a quantity  of  sugar  at  22  pounds  for  a dollar,  and  sold 
half  of  it  at  18  pounds  for  a dollar,  and  half  of  it  at  19  pounds  for  a dollar ; what 
per  cent,  did  he  gain  on  the  whole  ? 

17.  What  per  cent,  is  gained  by  buying  coal  at  $4.87J  per  ton  of  2210  lbs. 
and  selling  it  at  $5.50  per  ton  of  2000  lbs.  ? 

18.  What  per  cent,  profit  is  gained  by  buying  flour  at  $5.60  a barrel  and 
selling  it  at  85  cents  per  sack  of  24  lbs.  ? 

19.  What  per  cent,  profit  does  a druggist  make  by  buying  a powder  at  38 
cents  a pound  avoirdupois  and  selling  it  at  10  cents  an  ounce  apothecary? 

20.  What  per  cent,  profit  does  an  importer  gain  on  cloth  that  he  buys  for 
3.25  francs  a meter  and  sells  at  67 J cents  a yard? 


158 


PROFIT  AND  LOSS 


544.  To  find  the  cost  when  the  profit  or  loss  and  the  rate  are  given. 

Example. — If  $249.36  is  gained  by  selling  goods  at  4%  profit,  what  is  the 
cost  ? 

.04 )_  249.36  Or,  04  = of  base, 

$6234.  and  $249.36  X 25  = $6234. 

545.  Rule. — Divide  the  profit  or  loss  by  the  rate  expressed  decimally ; or 
reduce  rate  to  equivalent  fraction  and  solve  by  analysis. 

MENTAL  PROBLEMS 

546.  1.  A furniture  dealer  gained  $1  by  selling  chairs  at  a profit  of  20  per 
cent. ; what  did  the  chairs  cost  ? 

Solution. — A profit  of  20  per  cent,  is  equal  to  4 of  the  cost,  which  equals  $1  ; the  entire  cost 
equals  5 times  $1,  or  $5. 

2.  A man  sells  a lot  of  goods  at  $4  gain,  which  is  25  per  cent,  of  the  cost ; 
what  is  the  cost  ? 

3.  A loss  of  $20  is  sustained  by  selling  a horse  at  10  per  cent,  below  cost ; 
what  is  received  for  the  horse  ? 

If.  A boat  builder  gains  $20  on  the  sale  of  a boat,  which  is  equal  to  16f  per 
cent,  of  the  cost;  what  does  he  get  for  the  boat? 

5.  A carriage  is  sold  at  a gain  of  $25,  which  is  12J  per  cent,  of  the  cost; 
■what  is  the  selling  price  of  the  carriage? 

6.  A carriage  is  sold  at  a loss  of  $25,  which  is  12J  per  cent,  of  the  cost; 
what  is  the  selling  price  of  the  carriage? 

7.  I gain  10  per  cent.,  or  $15,  on  the  sale  of  a wratch  ; what  did  the  watch 
cost  me  ? 

8.  A jeweler  gained  $10  on  the  sale  of  a ring,  which  was  20  per  cent,  of  the 
cost;  what  was  the  ring  marked,  if  it  wras  marked  10  per  cent,  above  the  selling 
price  ? 

9.  A cow  was  sold  at  a loss  of  $6,  which  was  equal  to  5 per  cent,  of  its 
value  ; what  should  have  been  obtained  for  the  cow  so  as  to  neither  gain 
nor  lose  ? 

10.  A loss  of  50  cents  on  a barrel  of  flour  is  equivalent  to  a loss  of  121  per 
cent. ; what  is  the  flour  worth  a barrel  ? 

11.  A man  gained  $25  by  selling  a boat  at  25  per  cent,  above  its  cost ; wbat 
would  he  have  gained  by  selling  it  at  15  per  cent,  above  its  cost? 

12.  $60  less  was  asked  for  a piano  at  a discount  of  30  per  cent,  from  list ; 
what  was  the  list  price? 

13.  $32  is  4 per  cent,  of  a young  man’s  money,  whose  amount  is  equal  to  50 
per  cent,  of  his  companion’s  money  ; what  amount  have  both? 

Ilf.  A dealer  gains  20  per  cent.,  or  $12,  by  the  sale  of  a bicycle  ; what  would 
he  have  obtained  for  it  if  he  had  sold  it  at  a gain  of  $1S? 

15.  A watch  sold  at  $50  more  than  cost  gives  a gain  of  25  per  cent. ; what 
was  its  cost  ? 


PROFIT  AND  LOSS 


159 


16.  A watch  sold  at  $50  less  than  cost  gives  a loss  of  16-|  per  cent. ; what 
was  its  cost? 

17.  A desk  sold  at  a discount  of  $10,  or  16§  per  cent,  less  than  its  marked 
price,  realizes  cost ; what  was  its  cost  ? 

18.  A man  bought  a watch  for  $12  less  than  its  cost,  which  was  12J  per  cent, 
of  its  cost ; what  did  he  pay  for  the  watch  ? 

19.  Desks  marked  25  per  cent.,  or  $5  above  cost,  are  sold  at  cost;  what  is 
the  cost  of  a desk  ? 

20.  A bicycle  sold  at  $15  above  cost  gains  30  per  cent,  on  cost;  what  was 
its  cost  ? 


ORAL  EXERCISE 

547.  Find  the  cost  if 


1.  $40  = 5 % gain. 

2.  $25  = Ql%  gain. 

3.  $75  = 37J%  gain. 

J.  $84  = 12%  loss. 

5.  $125  = 62 J%  gain. 

6.  $96  = 8%  loss. 

7.  $225  = 62 \°/0  gain. 

8.  $1728  = 144%  gain. 

9.  $1100  = S3i%  gain. 


10.  $570  = 19%  loss. 

11.  $240  — S-g-%  gain. 

12.  $160  = 16§  % gain. 

13.  $120  = 12J%  ioss. 
Ilf..  $300  = 33^-%  gain. 

15.  $150  = 15  % loss. 

16.  $10  = 34%  gain. 

17.  $125  = 25%  gain. 

18.  $200  = 66§  % gain 


WRITTEN  PROBLEMS 

548.  1.  By  selling  goods  at  an  advance  of  16f%,  a merchant  gained 
$324.68 ; what  did  the  goods  cost  him  ? 

2.  James  Clark  bought  a carriage  and  sold  it  at  a loss  of  27J%.  With  the 
amount  received  he  bought  another  carriage,  which  he  sold  at  a profit  of  30%, 
gaining  $27  on  this  sale.  What  did  he  pay  for  the  first  carriage? 

3.  Sold  a property  for  $287.17  less  than  it  cost,  losing  thereby  13%.  What 
was  the  cost  ? 

J.  A dealer  sold  a quantity  of  goods,  4 at  a profit  of  20%  and  the  other  \ 
at  a loss  of  18%.  His  net  gain  was  $322.  Find  the  cost  of  the  goods. 

5.  Brown  sold  a cargo  of  lumber  at  a profit  of  29%.  47  per  cent,  of  his 
gain  was  $4211.67.  How  much  did  he  pay  for  the  cargo? 

6.  Goods  sold  at  an  advance  of  $350,  realized  a gain  of  35%  ; what  did  the 
goods  cost  ? 

7.  A house  was  sold  for  $270  less  than  its  value,  at  a loss  of  3%  ; what  was 
the  price  obtained  for  it? 

8.  A property  wTas  sold  at  a gain  of  $550,  which  wms  IS  % of  its  cost ; 
what  was  obtained  for  it  ? 

9.  Merchandise  sold  during  a year  showed  an  average  gain  of  30%  ; if  the 
amount  gained  was  $3250,  what  was  the  amount  sold? 

10.  A corporation  set  aside  a dividend  fund  of  $12500,  which  was  24  % of 
the  capital  stock  ; what  was  the  capital  stock  ? 


160 


PROFIT  AND  LOSS 


549.  To  find  the  cost  when  the  selling  price  and  the  rate  per  cent,  of 
profit  or  loss  are  given. 

Example. — By  selling  goods  for  $5239.36,  12%  was  gained.  What  did  the 
goods  cost  ? 

46  78 
1.12  ) 5239.36 
448 
759 
672 

873 

784 

gQg  Result,  $4678. 

550.  Rule. — Divide  the  selling  price  by  1 {100%)  plus  the  rate  of  gain,  or  by  1 
minus  the  rate  of  loss. 

MENTAL  PROBLEMS 

551.  1.  A horse  sold  at  $110  yields  a gain  of  10%  ; what  was  the  cost  of 
the  horse  ? 

Solution. — A gain  of  10  per  cent,  is  a gain  of  T\T  of  the  cost ; the  cost  and  TU  of  the  cost  or  -J-J 
of  the  cost  equals  $110  ; W °f  the  cost  equals  of  $110,  or  $10,  and  the  cost  is  §100. 

1.  By  selling  a horse  at  $90,  a loss  of  10  per  cent,  is  sustained;  what  was 
the  cost  of  the  horse? 

3.  By  selling  merchandise  for  $60,  a gain  of  20  per  cent,  is  realized ; what 
is  the  cost? 

Ip.  Chairs  sold  for  $30  a dozen  realize  a gain  of  20  per  cent.;  what  did 
they  cost  ? 

5.  Coffee  sold  at  35  cents  a pound  yields  a gain  of  16§  per  cent. ; what  was 
its  cost  per  pound  ? 

6.  33J  per  cent,  is  gained  by  selling  a wheel  for  $40  ; what  did  it  cost? 

7.  Furniture  inventoried  at  $180  shows  a depreciation  of  10  per  cent.; 
what  did  it  cost? 

8.  $105  in  money  returned  to  a lender  includes  5 per  cent,  interest;  what 
was  the  amount  loaned  ? 

9.  Land  sold  at  $66  an  acre  shows  again  of  10  percent.;  what  was  its  cost? 

10.  $160  is  obtained  for  a carriage,  which  is  a loss  of  20  per  cent. ; what  was 

the  cost  ? 


ORAL  EXERCISE 


552.  Find  the  cost  if 

1.  $600  sale  nets  20%  gain. 

2.  $600  sale  nets  20  % loss. 

3 $350  sale  nets  16§  % gain 
4~  $350  sale  nets  16f%  loss. 

5.  $950  sale  nets  5%  loss. 


6.  $1190  sale  nets  19%  gain 

7.  $264  sale  nets  10%  gain. 

8.  $333  sale  nets  11%  gain. 

9.  $111  sale  nets  200%  gain. 
10  $315  sale  nets  25%  loss. 


PROFIT  AND  LOSS 


161 


WRITTEN  PROBLEMS 

553.  1.  Two  horses  were  sold  for  $100  each,  one  at  a gain  of  20%,  the  other 
at  a loss  of  20%.  Howr  much  was  gained  or  lost  on  both? 

2.  A sold  a house  to  B for  33J%  less  than  it  cost  him.  B sold  it  to  C for 
25  % more  than  he  paid  for  it.  C paid  $2500  for  the  house.  What  did  it  cost  A ? 

3.  Sold  a lot  of  merchandise  at  a gain  of  21|-%,  and  with  the  sum  received 
purchased  another  lot  which  1 sold  for  $4535.17,  at  a loss  of  3%.  Find  the  cost 
of  the  first  lot. 

A A jobber  offered  a lot  of  goods  at  18%  more  than  they  cost  him,  but 
afterward  sold  them  for  $1237.76,  which  was  7f%  less  than  his  first  offer.  How 
much  had  the  jobber  paid  for  the  goods?  What  rate  of  profit  did  he  make 
on  them  ? 

5.  If  goods  which  were  marked  31  % above  cost,  were  afterward  sold  for 
90%  of  the  marked  price,  yielding  a profit  of  $9392.89,  what  was  their  cost? 

6.  A merchant’s  capital  at  the  end  of  a business  year  is  $16619.80  ; during 
the  year  he  lost  8 % ; what  was  his  capital  at  the  beginning  of  the  year  ? 

7.  A shipment  realized  $6887.50,  at  a loss  of  5%  ; what  was  the  value  of 
the  shipment? 

8.  Goods  are  insured  for  $8287.50,  which  is  2J%  less  than  their  value; 
what  are  the  goods  worth  ? 

9.  An  agent  obtained  $956.25  for  goods,  which  was  a gain  of  12|%  ; what 
was  the  cost  of  the  goods? 

10.  Goods  marked  $1620  were  sold  for  $1350,  wrhich  "was  a loss  of  10%  ; how 
much  were  they  marked  above  cost? 

REVIEW  PROBLEMS  IN  PROFIT  AND  LOSS 

554.  1.  A hardware  dealer  buys  locks  at  $4.80  per  dozen,  list  price,  less  25  % 
and  20%,  and  sells  at  the  same  list  price  less  20%  and  16f  %.  What  is  his  gain 
on  125  dozen,  and  his  gain  per  cent.  ? 

2.  A clothier  gains  25%  by  selling  cloth  at  $5  per  yard,  but  a bale  of  80 
yards  being  damaged,  he  has  to  reduce  the  selling  price  10%.  What  is  his 
profit  on  this  damaged  bale,  and  his  gain  per  cent.? 

3.  If  green  coffee  costs  25  cents  a pound,  and  1 cent  a pound  for  roasting, 
for  what  must  a pound  of  roasted  coffee  sell  to  gain  16§  %,  if  green  coffee  shrinks 
15  per  cent,  in  roasting  ? 

A A drover  sold  a horse  for  $180  and  lost  25%  ; with  this  money  he  bought 
another  horse,  which  he  sold  at  a gain  of  25%.  What  was  his  gain  or  loss  on 
the  transaction  and  gain  or  loss  per  cent.? 

5.  A man  bought  a horse  and  carriage  for  $450  paying  25%  more  for  the 
horse  than  for  the  carnage.  He  sold  the  horse  at  a gain  of  40  %,  and  the  carriage 
at  a loss  of  30%.  What  did  he  gain  on  the  transaction? 


162 


PROFIT  AND  LOSS 


6.  A merchant  bought  dry  goods  for  $6000;  he  sold  20%  of  them  at  a 
gain  of  10%  ; 40%  of  them  at  a gain  of  20%  ; 40%  of  the  remainder  at  a gain  of 
5%,  and  the  remainder  at  cost.  What  did  he  gain  or  lose? 

7.  My  retail  price  of  fans  is  $2.50  each ; by  selling  them  at  this  price  I 
would  gain  25%.  I sell  the  same  fans  wholesale,  at  $40  per  dozen,  less  124% 
and  20%.  Do  I gain  or  lose  at  the  wholesale  price,  and  how  much  a fan  ? 

8.  A merchant  marked  his  goods  60%  above  cost.  He  gave  one  of  his 
customers  a discount  of  15%  off  the  marked  price;  what  was  his  gain  on  $6.80 
received  from  that  customer  ? 

9.  A merchant  sold  20  % of  an  invoice  at  30  % profit ; 25  % of  the  remainder 
at  20%  profit;  what  was  his  total  gain,  if  the  cost  of  the  goods  unsold  is  $3600? 

10.  How  should  goods  be  marked  that  cost  $5  so  as  to  offer  a discount  of 
20%  and  10%  and  still  make  a profit  of  25%  ? 

11.  A man  bought  a lot  of  apples  and  lost  25%  of  them.  What  must  be  his 
per  cent,  of  gain  on  the  remainder  to  net  a gain  of  20%  on  the  cost  of  the  lot? 

12.  A man  sold  a house  at  30%  profit,  and  with  this  money  bought 
another  house  which  he  sold  at  25%  profit.  What  did  he  pay  for  each  house  if 
the  total  gain  was  $2500  ? 

13.  A merchant  buys  cloth  at  $2  50  a yard.  How  should  it  be  marked  to 
gain  25%  if  the  merchant  loses  5%  of  his  sales  in  bad  debts? 

Ij, l.  I sold  a lot  for  $102.25  more  than  cost  and  gained  5%.  I sold  another 
lot  which  cost  the  same  for  $3000.  What  was  my  gain  per  cent,  on  the  second 
sale  ? 

15.  The  retail  price  of  an  article  was  $262.50,  which  was  25%  more  than  cost 
to  the  retailer.  The  retailer  bought  of  a jobber  who  sold  it  at  a gain  of  20%  above 
the  manufacturer’s  selling  price  and  the  manufacturer  made  a profit  of  16f%. 
What  did  it  cost  to  manufacture  this  article,  and  what  was  each  one’s  profit? 

16.  A bought  80  yards  of  broadcloth  at  $3.40  per  yard,  and  74  yards  of 
cassimere  at  $2.50  per  yard.  He  sold  the  cassimere  at  a loss  of  20%.  What 
should  he  ask  per  yard  for  the  broadcloth  to  net  a gain  of  25%  on  the  cost 
of  both  ? 

17.  A sold  a wagon  to  B and  lost  25%.  B sold  it  to  C at  a loss  of  5%  ; C 
spent  $80  for  repairs  and  sold  it  for  $472.25  gaining  20%.  What  did  the 
wagon  cost  A,  B and  C respectively  ? 

18.  A real  estate  dealer  bought  three  houses  at  $1800  each,  and  sold  them  at 
a profit  of  8%,  12%  and  15%  respectively.  What  was  his  gain  on  the  three 
houses  ? 

19.  A real  estate  agent  sold  three  houses  for  $2400  each,  at  a gain  of  20%, 
124%  and  15%  respectively.  What  wTas  his  entire  gain? 

20.  How  much  should  be  asked  for  coffee  which  costs  18  cents  a pound  to 
gain  10%,  allowing  10%  for  loss  in  roasting? 


INVOICE  EXTENSION  AND  TRADE  DISCOUNT 


555.  Invoice  Extension,  or  Bill  Work,  consists  in  multiplying  the  items 
of  a bill  or  invoice  by  their  respective  prices  and  placing  the  results  in  the 
money-column  to  the  right. 

Note. — It  is  not  necessary  that  students  be  advanced  in  arithmetic  to  take  the  drill  in  invoice 
extension.  Any  student  sufficiently  advanced  to  take  a business  course  can,  in  a short  time,  be  taught 
to  make  the  extensions,  and  we  advise  that  it  be  made  a matter  of  daily  practise.  For  this  purpose  we 
recommend  bills  of  five  items  for  beginners,  the  quantities  and  prices  to  contain  fractions  frequently 
met  with  in  business,  as  £,  J,  £,  §,  f,  f,  -§,  f,  etc.,  together  with  a series  of  three 
trade  discounts  on  each  item.  It  is  well  to  have  a number  of  such  invoices  which  should  always  be 
dictated  and  the  work  timed.  A bill  of  five  items  should  be  correctly  extended  and  the  discounts  taken 
off  in  from  15  to  20  minutes,  the  latter  being  the  proper  time-limit  for  a class.  Beginners  are 
excused  from  taking  off  the  discounts  until  they  have  acquired  the  ability  to  make  the  gross  extensions 
within  this  time-limit,  when  they  should  begin  taking  off  the  easiest  discounts.  More  advanced  students 
may  be  given  bills  of  ten  items  which  should  be  extended  and  discounted  in  from  25  to  40  minutes.  A 
specimen  bill  such  as  is  used  in  this  work  is  given  on  page  167. 

556.  A discount  is  any  deduction  from  the  face  of  a bill  or  debt.  It  is 
usually  reckoned  as  a certain  rate  per  cent. 

557.  Commercial  discounts  are  of  two  kinds : (1)  Trade  discounts,  which 
are  deductions  from  the  fixed  or  list  price  of  goods,  allowed  to  the  “trade” — 
(those  in  the  same  line  of  business)  by  manufacturers,  jobbers  and  wholesalers 
(2)  Cash  discounts — deductions  for  immediate  payment,  made  from  the  net 
amount  of  a bill  for  payment  within  a definite  time. 

It  will  thus  be  seen  that  trade  discounts  are  absolute,  while  cash  and  time  dis- 
counts are  conditional — the  first  being  deducted  from  the  list  price  when  the 
goods  are  billed,  and  the  others  from  the  amount  of  the  bill  when  payment  is 
made  in  accordance  with  the  expressed  conditions. 

558.  The  list  price  is  also  called  the  marked  price,  asking  price,  offer- 
ing price,  or  gross  price  ; and  the  price  after  the  trade  discount  has  been 
deducted,  the  net  price  or  simply  the  “ net.” 

559.  Business  men  usually  announce  their  terms  on  their  billheads ; as, 

“ Terms  : Net  3 months,  or  5%  off  for  cash  ” ; “ Terms : Net  60  days,  or  2%  off 

in  10  days.” 

560.  Trade  discounts  arise  principally  from  two  different  causes: 

1.  In  many  lines  of  business  it  is  customary  for  merchants  to  issue  cata- 
logues and  price-lists  of  their  goods,  with  the  different  articles  listed  at  fixed 
prices  higher  than  the  actual  selling  prices.  They  then  issue  discount  sheets 
from  time  to  time,  varying  the  discounts  as  the  market  prices  change,  instead  of 
altering  the  list  prices. 

£.  In  almost  all  lines  of  business,  the  larger  quantity  a dealer  buys  at  one 
time  the  lower  price  he  can  get ; and  in  many  cases  this  concession  is  made- 
in  the  form  of  an  extra  discount. 


163 


164 


TRADE  DISCOUNT 


561.  A series  of  discounts  consists  of  two  or  more  discounts  to  be  taken  off 
successively,  the  first  discount  being  reckoned  on  the  list  price,  the  second  on  the 
remainder  left  after  subtracting  the  first,  the  next  on  the  remainder  left  after 
subtracting  the  second,  and  so  on. 

Note. — Observe  that  $100  less  20%  is  $80;  but  $100  less  10%  aud  10%  is  $81.  $100  less  40%  is 

$00;  but  $100  less  20%  and  20%  is  $64. 

562.  To  find  the  net  amount  of  a bill. 

Example  1. — Find  the  net  amount  of  a bill  of  $467.50,  subject  to  discounts  of 
7%,  3%  and  2%. 


7 

3 

2 


// 3 7.3-0 
3 2.7  2 


z/3  7-7  7 
/ 3.0  2 


<77  f.7*/ 
7.23 


44/  3.  3 / 


Explanation. — Multiply  by  7,  reject  two  right-hand  figures 
and  subtract;  multiply  the  remainder  by  3 and  reject  the  two  right- 
hand  figures;  multiply  the  second  remainder  by  2,  rejecting  and  sub- 
tracting as  before.  The  remainder  is  the  net  amount. 

The  difference  between  the  amount  of  the  bill  and  the  net 
amount  is  the  total  discount. 


563.  Rule. — Deduct  the  first  discount  from  the  list  price,  and  each  subsequent 
discount  from  the  respective  remainders. 

Note. — In  taking  off  discounts,  usage  regarding  fractions  of  a cent  differs.  For  uniformity, 
students  should  drop  all  such  fractions. 

Before  beginning  to  discount,  add  two  ciphers  for  cents  to  sums  expressed  in  dollars  only. 

Example  2 — Find  the  net  amount  of  $467.50  less  25%,  14-f- % and  SJ%. 


7%T 

/ //V 

3 3// 

r/j 


73  7.  so 
/ / 3.  7 7 
3 S 0.  3 3 
,7  O.  O 7 
3 0 O.S7 
2 S.  O 7 
27  S.SO 


^esT- 


Explanation. — Since  25%  equals  4,  divide  by  4 for  the 
first  discount;  divide  the  remainder  by  7,  since  144%  equals  4; 
divide  what  now  remains  by  12,  since  8J%  equals  TV.  The 
remainder  is  the  net  amount. 


Example  3. — Find  the  net  result  of  $467.50  less  22-|%,  624%  and  18J  %. 


3 2 '/is 

/ <23/ 

/ 7 /v 


447 

/ O 3.77 
3 3 3,32 


<?  J e?3S~Q  o 

rj  / r/ r/o 


227.23 

/ 3 3.3  3 y-)  17  otfo  r 

2 S.S  3 4JLL0Z2J- 

/ / 0 , 7 


Explanation. — 22|%  equals  f,  hence 
multiply  by  2 (placing  the  product  to  the  right), 
and  divide  it  by  9 for  the  discount;  since  621  % 
equals  f,  multiply  the  remainder  by  5 (placing  the 
product  to  the  right  as  before),  and  divide  it  by 
8;  since  18|%  equals  T\,  multiply  the  remainder 
by  3 aud  divide  by  16,  or  by  4 twice.  The 
remainder  is  the  net  amount  desired. 


Note. — In  discounts  requiring  two  or  more  operations,  place  to  the  right  all  results  except 
the  last. 


TRADE  DISCOUNT 


165 


Example  4. — Find  the  net  result  of  $467.50  less  2| -%,  3J%  and  6J%. 


J2^ 

t>'/¥ 


4 3 7. SO 
/ / .3  s 
4 ss.  2 2 
/ s.  / 4 
44  0.  3 3 

2 7.43 
4 / 3 . / 0 


yjjjjLJJL 


Explanation. — 2£%  equals  4 ; divide  by  4 
(writting  quotient  under,  but  one  place  to  the  right); 
since  3 J % equals  4 divide  the  remainder  by  3 ( writing 
quotient  under,  but  one  place  to  the  right);  for  6J-% 
(J5)  divide  the  remainder  by  4,  placing  the  entire 
quotient  to  the  right;  divide  this  quotient  by  4 for  the 
discount  ; the  second  quotient  subtracted  from  the 
second  remainder  leaves  the  net  result. 


Example  5. — Find  the  net  result  of  $467.50  less  15%,  74%  and  6f  %. 


/s 

77 

3 ^3 


4 & 7.  SO 
7 0./  4 


3 4 7-32 
2 4.  ro 


3 3 '7.sr 
2 4.  SO 
3 4 3.0  2 


x) /y  0 2.J-# 

¥ j / / ^-2./y 

3)  73S/  $ 


Explanation. — 15%  equals  4 hence  mul- 
tiply by  3,  and,  after  rejecting  right-hand  figure,, 
divide  the  product  by  2 ; multiply  the  remainder  by 

3,  and,  after  rejecting  the  right-hand  figure,  divide  by 

4,  since  7\  % equals  4>  multiply  the  remainder  by 
2,  and  divide  the  product,  exclusive  of  right-hand 
figure,  by  3,  because  6j%  equals  4-  The  remainder 
is  the  result. 


Example  6. — Find  the  net  result  of  $467.50  less  87J%,  75%  and  50%. 


77 

7 s 

SO 


437,40 

S2.  43 
/ 44  0 
7 • 3 0 syz^.4' 


Explanation. — If  874-%[be  taken  away,  only  12£%  is  left, 
hence^divide  by  8j;  since  75%  discount  leaves  only  25%,  divide 
the  result  by  4 ; likewise,  50%  off  leaves  50%,  hence  divide  by  2 ;: 
the  last  result  is  the  net. 


Example  7. — Find  the  net  result'of  $832  less  70%,  60%  and  30%. 


7o 
3 0 

3 0 


23  2.00 


247S 


O 


7 7 . 24 


3 7 ’ 2 2 


Explanation.— 70%  taken  off  leaves  30%  ; multiply  by 
3 and  reject  one  figure  for  the  net  after  this  discount  ; 60%  off 
leaves  40%,  hence  multiply  result  by  4 and  reject  one  figure  for 
the  net  after  the  second  discount  ; 30%  off  leaves  70%,  therefore 
multiply  former  result  by  7,  rejecting  one  figure.  The  product 
is  the  net  result  of  the  bill.  The  same  result  can  be  obtained  by 
multiplying  each  decimally. 


166 


TRADE  DISCOUNT 


TABLE  OF  SHORT  METHODS 


564.  The  short  methods  below  apply  to  sums  in  dollars  and  cents  ; for  sums 
in  dollars  only,  the  student  should  annex  ciphers  to  avoid  falling  into  the 
error  of  rejecting  fractions  of  a dollar.  Operations  should  be  performed  mentally, 
if  possible. 

1.  To  get  2%,  3%,  4%,  5%,  6%,  7%,  or  any  discount  not  an  easy  aliquot 
part  of  100%,  multiply  and  reject  two  right-hand  figures. 

2.  Toget50%,33i%,25%,20%,16|%,14f%,m%,lli%,10%,91ir%,8i%. 

Divide  by  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10)  (11)  (12) 

S.  To  get  22**  (|),  37i%  (|),  62**  (*),  44**  (*),  18**  (A). 

Multiply  by  2 ande-9  ; 3 and-4-8  ; 5 and-t-8  ; 4 and^-9  ; 3 ande-16,  or  4 twice. 

T To  get  5 % (i),  divide  by  2 and  set  quotient  one  place  to  the  right. 

To  get  3J%  (-g1^),  divide  by  3 and  set  quotient  one  place  to  the  right. 

To  get  2J%  (^-),  divide  by  4 and  set  quotient  one  place  to  the  right. 

To  get  (re),  divide  by  16,  or  by  4 twice. 

5.  To  get  6f  % (¥2-q),  multiply  by  2,  reject  right-hand  figure  and  divide  by  3. 
To  get  multiply  by  3,  reject  right-hand  figure  and  divide  by  4. 

To  get  13j-%  multiply  by  4,  reject  right-hand  figure  and  divide  by  3. 
To  get  15  % (yq),  multiply  by  3,  reject  right-hand  figure  and  divide  by  2. 
To  get  35  % (-^o),  multiply  by  7,  reject  right-hand  figure  and  divide  by  2. 


6.  To  get  net  results  if  discounts  are  50%  (4),  66f  % (§),  75%  (f),  80%  (f), 
■%  (f),  85f%  (f),  87 \%  (|),  90%  (gSg),  which  leave  but 


A 1 A 1 1 1 1 1 
2)  3)  i)  5)  6’  T>  8)  1 0 > 


divide  the  sum  by  2,  3,  4,  5,  etc. 


7.  To  get  30%,  40%,  60%,  70%,  multiply  by  left-hand  figure  (3,  4,  6,  7)  and 
reject  right-hand  figure  in  product ; or,  to  get  the  “ net,”  use  as  a multiplier  the 
left-hand  figure  of  the  difference  between  the  rate  and  100%. 


Note. — Examples,  corresponding  in  number  to  the  above  groups  and  illustrating  how  the  work 
shall  be  done,  are  found  on  the  preceding  pages. 

8.  A series  may  have  an  easjr  equivalent;  as,  40 % , 33 J- % and  25%  equals 
'70%  off;  33J%,  25%  and  20%  equals  60%  off. 


Example  S. — Find  the  net  result  of  $652.70  less  40%,  33J%  and  25%. 


/ 0 0°/° 
2 0 

33/3 

(0  0 / 
2 O 

2S 

2 0°/° 

3 0 


C>S2.70 
/ ^ <2.  2 / 


Explanation.— 40  % off  100  leaves  60  % ; 
334%  (i)  off  60%  leaves  40%;  25%  (1)  off  40% 
leaves  30%  ; 30%  of  $652.70  equals  $195.81. 


TRADE  DISCOUNT 


167 


565.  Form  of  invoice  showing  extensions,  discounts  and  net 
extensions. 


Note. — The  small  figures  at  the  right-hand  in  the  quantity  and  price  are  fourths  of  a yard  and 
fourths  of  a cent,  the  numerator  alone  being  written.  All  other  fractions  have  both  numerator  and 
denominator  expressed. 

The  operations  of  items  1 and  2 illustrate  the  method  and  form  to  be 
followed. 


/ 0 


S 


2 ' 


2 

/. 


3'A 


/ 3 / S 
2 3 y A 
2 t,  3 


'S'  - 


S/  */.  6 S 
S / .4  6 


/ 3 
4 

L. 


4 6>  3./  f 
2 3./ S 


4 4 0.04 

/ / .0  o 


4 2 y • Q 4 ^y^e4- 


Operation  1 

(1)  263X1-95. 

(2)  | of  263=131  j. 

(3)  i of  195=48|. 

(4)  i of  i=b 

(5)  i+f+l=lf. °rlc. 


Operation  2 

(1)  357X2.18. 

(2)  f of  357=267|. 

(3)  f of  218=145h 

(4)  f of  f=xV 

(5) t62+1+|— TTi0r2c. 


-2dT 


/ 2- 


6' 


3 S 
2./ 


2 fS  6 

3 sy 
7/4 

2 6>73//  ~ 


2 >43  6, 
'4  ) /o  7 / 


6>/  — h 
7/2 


/ 4 


2 


= f 
= 4, 
42-. 
/ z 


y 7 2.4  0 
/ 4 s.  k 0 


sf6.ro 

7 3.3  s 


S / 3.4  S 
3 2.0  4 


4)  / 2.?3& 


4 f / • 3 /a 


Note  — Less  than  5 cent  in  a final  result  is  rejected  ; more  than  1 cent  is  counted  a cent. 


168 


TRADE  DISCOUNT 


566.  The  multiplications  may  often  be  shortened  by  the  use  of  aliquots,  as 
explained  under  Quantity,  Price  and  Cost,  (page  90)  and  as  illustrated  by  items 
4 and  5 in  the  model  invoice  on  preceding  page. 


V6  2, 

37s 

/ 2-// 

/ 2S 

/ / 6 

SO 

2 f 7. 

so  =, 

s 

14  3 7 

so  = 

3 S 

f 3 JS 

//  40  / 

stss 

Note. — It  should  be  noted  here  (1)  that  while  the  items  given  for  practise  are  from  the 
dry  goods  business,  the  discounts  are,  for  the  most  part,  those  met  with  in  the  wholesale  hardware, 
wooden  ware,  and  similar  businesses,  where  single  discounts  will  sometimes  run  as  high  as  90%  or 
more  ; (2)  that  while  three  discounts  are  given  here  for  uniform  practise,  in  business  the  number  will 
vary  from  one  to  a series  of  five  or  more  ; (3)  that  two  or  more  items  in  a bill  may  be  discounted  at 
the  same  rate,  in  which  case  the  gross  extensions  are  added  and  the  discounts  taken  oft'  at  one  time. 


MENTAL  PROBLEMS 

567.  Example. — What  is  the  net  price  of  an  article  listed  at  $8,  25  per 
cent,  off? 

Solution.— 25  per  cent,  off  is  f off  or  $2  off;  $8  less  $2  is  $6  net. 

2.  What  is  the  net  wholesale  price  of  tables  listed  at  $15  less  20  per  cent.? 

3.  What  is  the  net  price  of  chairs  listed  at  $15  a dozen  less  33 J per  cent.? 

If..  What  is  the  list  price  of  goods  sold  at  $21  net,  discount  124  per  cent.? 

5.  A merchant  buys  at  wholesale  for  $10,  which  is  16f  per  cent,  off  list* 
what  sum  would  he  gain  by  selling  retail  at  list? 

6.  What  is  the  net  cost  of  goods  listed  at  $150,  sold  20  and  10  per  cent,  off? 

7.  State  the  net  cost  of  merchandise  listed  at  $200,  25  and  10  per  cent,  off- 

8.  What  is  the  retail  price  of  a piano  listed  at  $450,  sold  less  60  per  cent.? 

9.  A book  retailing  at  list  price  was  bought  at  $3,  less  20  per  cent,  and  10 
per  cent.;  how  much  was  gained  on  the  book  ? 

10.  A buyer  bought  hats  at  $60  a dozen  less  33£  per  cent,  and  25  per  cent,, 
and  was  given  an  additional  discount  of  5 per  cent,  for  cash ; find  net  cost  to 
the  buyer. 

11.  How  much  better  is  a single  discount  of  30  per  cent,  than  a series  of  20 
per  cent,  and  10  per  cent,  off? 

12.  What  is  the  difference  in  favor  of  a single  discount  of  40  per  cent.,  over 
a series  of  20  per  cent,  and  20  per  cent.? 

13.  Which  would  you  prefer — to  buy  goods  at  40  per  cent,  off  list,  or  to  buy 
them  25  per  cent,  and  20  per  cent,  off? 

Ilf.  Which  is  the  better  and  how  much — list  less  10  per  cent,  and  20  per 
cent.,  or  list  less  30  per  cent,  ? 

15.  Which  is  the  better — goods  bought  at  30  per  cent,  and  20  per  cent,  off, 
or  at  25  per  cent.,  20  per  cent,  and  5 per  cent,  off? 


TRADE  DISCOUNT 


169 


ORAL  PROBLEMS 

568.  1 ■ What  per  cent,  of  the  list  price  is  the  net  selling  price  of  goods  sold 


at  20%,  12 4%  and  10%  off? 

2.  At  50%  and  10%  off? 

3.  At  30%  and  10%  off? 

If.  At  40%  and  25%  off? 

5.  At  25%,  20%  and  10%  off? 


6.  At  50%,  40%  and  20%  off? 

7.  At  20%,  10%  and  84%  off? 

8.  At  30%  and  10%  off? 

9.  At  40%,  16$  % and  10%  off? 

10.  At  374%  and  20%  off? 


WRITTEN  EXERCISE 


569.  Find  the  net  amount  of 

1.  $120  less  25%,  10%  and  10%. 

2.  $372.80  less 30%,  124%  and  2%, 

3.  $732.17  less  334%  and  3%. 

If.  $84.60  less  10%,  4%  and  2%. 

5.  $497.83  less  40%,  224  % and  1% 


6.  $1200.62  less  17%  and  7%. 

7.  $696.34  less  75  %,  324% and  34% 

8.  $754.55  less  20%,  10%  and  6f%. 

9.  $3651.48  less 60%  13%  and  5%. 

10.  $587.30  less  80%  and  114%. 


WRITTEN  PROBLEMS 

570.  1.  A bill  of  goods  amounting  to  $347.60  is  sold  May  10,  subject  to 
discounts  of  30%,  10%  and  5%,  terms,  net  60  days  ; or  2%  off  if  paid  in  ten  days. 
How  much  will  pay  the  bill  May  18? 

2.  How  much  must  be  the  list  price  of  goods  subject  to  discounts  of  20% 
and  10%  in  order  to  net  $288? 

3.  What  is  the  difference  on  a bill  of  $582.50  between  a discount  of  45% 
and  discounts  of  25%,  10%  and  10%  ? 

If.  Find  the  net  cost  of  40  doz.  locks  at  $3.10,  less  20%  and  5%;  56  doz. 
hinges  at  $2.20,  less  334%;  90  doz.  bolts  at  $2.55,  less  60%,  10%  and  10%;  with 
an  allowance  of  2%  for  cash  payment. 

5.  What  must  be  the  list  price  of  goods  that  cost  $97.80  in  order  to  gain 
20%,  if  they  are  sold  at  25%,  10%  and  2%  off? 

6.  What  is  the  net  cost  of  1286  yards  of  cloth  at  874  cents  per  yard,  less 
10%  and  5%,  with  a discount  of  2%  for  cash  ; case  and  packing  $2.25? 

7.  If  goods  are  bought  at  discounts  of  25%,  20%  and  5%  from  list  price, 
and  sold  at  20%  and  10%  from  list,  what  is  the  gain  per  cent.? 

8.  How  much  per  dozen  is  gained  by  buying  brushes  at  $5  per  dozen,  less 
30%,  10%  and  10%,  and  selling  them  at  35  cents  each? 

9.  What  advance  on  cost  is  necessary  in  order  to  give  a discount  of  25% 
and  still  make  a profit  of  334%  ? 

10.  A jobber  buys  5 doz.  lawn  mowers  at  $50  per  doz.,  less  40%,  124%  and 
3%,  and  sells  them  to  a retailer  at  a price  that  gives  him  a net  profit  of  $52.65 
on  the  lot.  What  price  per  dozen  does  the  retailer  pay,  and  for  how  much  apiece 
must  be  sell  the  mowers  to  gain  25%  ? 


170 


TRADE  DISCOUNT 


Exercise  in  Invoice  Extension  and  Discounting 


1. 

571. 

Items. 

3522  yds. 

Prices. 

$2.622 

Discounts. 

9-6-2 

2. 

532f 

2.371 

8-6-4 

3. 

389 3 

a 

3.22 2 

11—7—3 

1 

516f 

a 

1-33* 

6—4—1 

5. 

1932 

a 

1.16| 

8—5—2 

6. 

2722 

a 

1.122 

20-10-5 

7. 

266* 

a 

2.33* 

25— 122— 10 

8. 

2033 

a 

2183 

33i_16|-8i 

9. 

1922 

a 

2.872 

20—10—10 

10. 

263 3 

a 

3.25 

50— 25— 122 

11. 

297f 

a 

1.372 

372— 183— 10 

12. 

3682 

a 

1.50 

221-111-5 

13. 

183f 

a 

1.622 

622 — 372 — 122 

n 

4863 

a 

2.16f 

441-221-111 

15. 

393* 

a 

2.33| 

372— is3— 142 

16. 

2183 

a 

1 .73 3 

10— 5-22 

17. 

3822 

a 

1.25 

10— 3i— 2 2 

18. 

238* 

It 

5.16| 

61—5—31 

19. 

1961 

a 

1.20 

61—  3i— 22 

20. 

188| 

a 

1.872 

5— 3i— 2 2 

21. 

303  3 

a 

2.813 

15— 72— 22 

22. 

361 1 

Ci 

1.322 

l3i_6|_3i 

23. 

245  2 

a 

2.1 22 

35— 15— 72 

21f. 

243f 

a 

2.66| 

15—72—62 

25. 

753f 

a 

1.75 

13i_6!_5 

26. 

6153 

a 

3.33* 

75—25—10 

27. 

1792 

a 

2.372 

872—122—61 

28. 

432* 

a 

4.183 

90—10—5 

29. 

357f 

a 

2.50 

66|-33J-3i 

30. 

247  2 

a 

5.66f 

80—50—20 

31. 

5952 

a 

1.172 

40—30—10 

32. 

125 

a 

7.562 

70—60—30 

33. 

3122 

a 

3.75 

60—40—20 

31  975 

a 

1.272 

30-10-3 

35. 

477 3 

a 

1.55 

70—40—30 

36. 

133* 

a 

4.632 

331—25—20 

37. 

250 

a 

2.95f 

25—20—161 

38. 

428f 

a 

2.622 

40—331-25 

39. 

333 3 

a 

3.331 

50— 20-122 

4.0.  750 

a 

1.47f 

60—331—25 

Items. 

Prices. 

DLscounts. 

41. 

261 1 yds. 

$2.06x 

5—10—20 

42. 

1922 

u 

2.872 

25— 122— 61 

43. 

203 3 

(( 

2.183 

33*— 16f— 8* 

44- 

191| 

a 

2.33i 

15— 7 2— 2 2 

45. 

225i 

a 

1.622 

3*—10—30 

46. 

1022 

a 

1.122 

20—10—5 

47. 

121 1 

a 

2.061 

10— 5— 2 2 

48. 

1333 

a 

1.33i 

15 — 10 — 5 

49. 

155f 

a 

1.372 

25— 12  2—6 1 

50. 

163f 

a 

1.622 

33*— 16|— 8* 

51. 

177| 

a 

1.872 

15 — 7 2 — 5 

52. 

191i 

a 

1.66f 

l0—3*—2 

53. 

221 1 

“ 

2.061 

5—10—20 

54- 

25 2 2 

a 

2.122 

61 — 122 — 2* 

55. 

2633 

a 

2.1 83 

15— 7 2— 2 2 

56. 

271i 

(( 

2.331 

66* — 33* — 16* 

57. 

283f 

u 

2.622 

30—15—5 

58. 

295| 

a 

2.66-1 

10 — 13* — 3* 

59. 

297i 

a 

2.872 

14* — 16* — 20 

60. 

303 3 

a 

3.372 

11*— 37 2— 50 

61. 

3682 

a 

2.16f 

25 — 15 — 5 

62. 

393 3 

a 

5.122 

5 2 2 2 2 

63. 

263J 

a 

4.183 

6—3—2 

64. 

567f 

a 

2.66| 

3*— 5— 10 

65. 

3893 

a 

1.331 

13*— 72— 122 

66. 

3732 

a 

1.95f 

50— 14*— 7 

67. 

737| 

a 

4.22 2 

44|— 122— 5 

68. 

149* 

a 

1-58* 

55*  13| — 3* 

69. 

2811 

u 

1-83* 

66* — 281 — 6* 

70. 

1932 

a 

2 43* 

1 1 * — 11* — 4 

71. 

387-| 

a 

1.93* 

83*— 16*— 10 

72. 

376 3 

a 

2.11* 

22|— 18|—6 1 

73. 

152f 

a 

2.43* 

75 — 50 — 20 

74. 

235| 

u 

4.33* 

90—20—25 

75. 

627-| 

a 

1-97* 

40—10—10 

76. 

1841 

a 

5.872 

5 q 1 q 

lTT  y 

77. 

846  3 

a 

2.58* 

55 — 12 — 6 

78. 

277f 

a 

1.52 2 

574 — 14* — 7 

79. 

3842 

a 

3.45f 

80—25—33* 

80. 

585f 

u 

6.933 

85 — 33* — 50 

COMMISSION  AND  BROKERAGE 


572.  Commission  or  brokerage  is  the  compensation  received  by  a commis- 
sion merchant  for  selling  goods,  or  by  an  agent  or  broker  for  either  buying  or  selling 
for  another. 

573.  Commissions  are  usually  reckoned  at  a certain  rate  per  cent,  of 
the  amount  sold  or  purchased. 

Note. — An  agent  receives  his  commission  on  the  gross  -proceeds  of  a sale,  or  on  the  prime  cost 
of  a purchase. 

574.  An  account  sales  is  a detailed  statement  rendered  by  a commission 
merchant  or  other  consignee  to  his  principal,  the  consignor,  showing  the  sales  of 
the  consignment,  the  incidental  charges  and  expenses  and  the  net  proceeds. 

The  net  proceeds  is  the  sum  left  after  the  commission  and  other  charges  have 
been  deducted. 

575.  An  account  purchase  is  a detailed  statement  rendered  by  an  agent  to 
his  principal,  showing  the  goods  bought,  prices  paid,  incidental  charges  and 
expenses,  and  the  gross  cost. 

576.  To  find  the  commission  when  the  amount  of  purchase  or  sale 
and  rate  of  commission  are  given. 

Example. — At  3%,  what  does  a commission  merchant  receive  for  selling 
$3275  worth  of  goods? 

$3275 

.03 

$98.25 

577.  Rule. — Multiply  the  gross  proceeds  of  the  sale,  or  the  prime  cost  of  the 
purchase,  by  the  rate  per  cent,  of  commission. 


MENTAL  PROBLEMS 

578.  1.  An  agent  receives  12J  per  cent,  commission  for  selling  sewing 
machines ; what  amount  would  he  receive  for  selling  one  for  $80? 

Solution. — 12£  per  cent,  commission  is£  of  the  amount  sold  ; J of  $80  is  $10,  the  amount  of 
commission. 

2.  What  amount  of  commission  would  be  received  for  selling  bicycles  for 
$75  at  a rate  of  33J  per  cent.  ? 

3.  An  agent  gets  a 40  per  cent,  commission  for  selling  books  by  subscrip- 
tion ; what  would  he  receive  on  an  art  work  sold  at  $30? 

A commission  merchant,  whose  rate  is  5 per  cent,  commission  and  2J 
per  cent,  guaranty,  makes  a sale  of  $200  worth  of  butter ; how  much  should  he 
retain  ? 


171 


172 


COMMISSION 


5.  An  agent  sells  wheels  at  a commission  of  30  per  cent. ; what  sum  would 
he  remit  to  his  principal  for  a wheel  sold  at  $80  ? 

6.  Grain  is  sold  at  a commission  of  2 per  cent. ; what  sum  should  be 
returned  on  a sale  of  $300  ? 

7.  An  agent’s  commission  is  3 per  cent,  for  buying  leather  ; what  would  be 
his  charge  on  an  invoice  worth  $500  ? 

8.  What  would  be  remitted  to  a consignor  for  an  invoice  of  4000  pounds 
of  sugar  at  4 cents  a pound,  less  agent’s  commission  of  3 per  cent.? 

9.  What  is  the  commission  at  5 per  cent,  on  goods  of  which  the  prime 
cost  is  $700  ? 

10.  If  the  prime  cost  is  $420  and  the  rate  of  commission  is  5 per  cent.,  what 
is  the  amount  of  commission  ? 

11.  A real  estate  agent  receives  5 per  cent,  for  collecting  a monthly  rental 
of  $30;  what  would  be  the  amount  received  by  him  in  a year  ? 

12.  A tax  collector’s  commission  is  2 per  cent. ; what  would  he  receive  on  a 
tax  of  $150? 

13.  What  amount  would  a broker  receive  on  the  sale  of  a $100  share  of  stock 
at  | per  cent,  brokerage  ? 

Ilf..  What  would  be  the  amount  remitted  by  a collector  who  has  collected 
an  account  of  $350  on  a commission  of  10  per  cent.? 

15.  What  should  I receive  at  a rate  of  5 per  cent,  for  collecting  25  per  cent, 
of  a claim  of  $800  ? 

16.  If  I collect  66§  per  cent,  of  a claim  of  $450  on  a commission  of  16§  per 
cent.,  what  amount  should  I turn  over? 

17.  What  are  the  net  proceeds  of  a sale  of  $300  worth  of  goods,  rate  of 
commission  5 per  cent.,  other  expenses  $10? 

18.  What  should  I receive  for  buying  $500  worth  of  produce  on  a commis- 
sion of  24  per  cent.?  How  much  should  my  principal  send  me  to  cover  purchase? 

19.  What  should  my  principal  send  me  to  cover  a purchase  of  $600,  commis- 
sion 3 per  cent,  and  2 per  cent,  for  guaranty  of  quality? 

20.  I gave  a collector  bills  to  the  amount  of  $250  to  collect ; at  5 per  cent, 
what  must  I pay  him  for  collecting  80  per  cent,  of  this  amount? 


WRITTEN  PROBLEMS 

579.  1.  A real  estate  broker  sold  a house  and  lot  for  $7245  ; what  was  his 
commission  at  2%  ? 

2.  At  brokerage,  how  much  must  I pay  my  broker  for  purchasing; 
$10000  worth  of  stock  ? 


COMMISSION 


173 


3.  Find  the  net  proceeds  of  the  following  account  sales  : 

Account  Sales 

Of  Merchandise  received  per  P.  It.  R.,  July  2,  1908,  from 
Sandusky  d:  Co.,  St.  Louis,  Mo.,  to  be  sold  on  their  account  and  risk. 


July 


7 

8 
10 

13 

14 


500  lbs.  Cheese  @$0.102 

250  “ Butter  .252 

75  doz.  Eggs  .142 

125  lbs.  Butter  .28 

475  “ Cheese  .11 2 

Charges. 

5.25  3.50 


Freight  and  Drayage 
Commission  on  Sales,  24% 

Your  net  proceeds 

Woodside  & Co. 

E.  and  O.  E. 

Philadelphia,  July  15,  1908. 


J.  Find  the  net  proceeds  of  the  following  account  sales  : 

Account  Sales 

Philadelphia,  Aug.  15,  1908. 


Sold  for  account  of  J.  D.  Tuckey  & Co., 

By  Snodgrass,  Murray  & Co. 


July 

2 

3 

9 

15 

642  yds.  English  Tweed 
879f  “ Broadcloth 
48J  “ Cheviot 
21331  “ Satinet 

@ $2.35 
5.62 2 
5.25 
.75 

Charges. 

6.25  2.25 

Freight  and  Cartage 

Commission  3% 

8 

50 

l 

Net  proceeds 

174 


COMMISSION 


5.  Find  the  total  cost  of  the  following  account  purchase  : 


Account  Purchase 

Bought  for  account  of  Wilson,  Brown  & Co., 


Chicago,  Sept.  1,  1908. 
By  William  R.  Adams. 


200 

bbls.  XX  Flour 

@ $2.25 

325 

“ Corn  Meal 

3.122 

150 

“ XXX  Flour 

6.75 

45 

“ Flour 

6.872 

25 

tons  Bran 

22.25 

Charges. 

Cartage 

Commission  3% 

Total  cost 

580.  To  find  the  gross  proceeds  or  prime  cost  when  the  net  proceeds 
or  gross  cost  and  rate  of  commission  are  given. 

Example  1. — My  agent  remitted  $2388.73,  proceeds  of  goods  sold  for  me  at 
1J%  commission  ; how  much  did  he  receive  for  the  goods  ? 


100  % Or  1.00 

H % -OH 

.981  % .98J 


2425  106 
.985  ) 2388.730 
1970 
4187 
3940 
2473 
1970 
5030 
4925 
1050 
985 
6500 


Or, 


.985  ) 2388.730  ( 2425.10 
1970 


5 

9“S7 


4187 

3940 

2473 

1970 


5030 

4925 

1050 

985 


650 

Result,  $2425.11. 


Example  2. — I remit  my  agent  $8733  with  which  to  purchase  wheat,  after 
deducting  his  commission  at  2J%.  How  many  bushels  can  he  buy  at  71  cents? 


100  % Or  1.00  8520  ' 1.025  ) 8733.000  ( 8520 


2i  % .021  1.025 ) 8733.000|  8200 

102i  ^ 1.02J  8200  5330 

5330  5125 

5125  2050 

2050  2050 

2050 


12000 
.71  ) 8520.00 
71 
142 
142 
000 

Result,  12000  bus. 


581.  Rule. — Divide  the  net  proceeds  by  1 (100%)  minus  the  rate  expressed 
decimally , or  the  gross  cost  by  1 (100%)  plus  the  rate. 


COMMISSION 


175 


MENTAL,  PROBLEMS 

582.  1.  The  gross  cost  of  a desk  is  $80  and  the  agent’s  commission  is  33 J 
per  cent.;  what  is  the  prime  cost? 

Solution. — The  prime  cost  plus  33J  per  cent,  of  the  prime  cost  isf  of  the  prime  cost  or  the 
gross  cost  of  $80  ; J of  the  prime  cost  is  £ of  $80  or  $20,  and  the  prime  cost  is  $60. 

2.  The  cost  of  an  organ  was  $100,  which  included  the  agent’s  commission 
of  25  per  cent.;  what  was  the  price  to  the  agent  ? 

3.  What  were  the  gross  proceeds  of  a sale  which  yielded  $80  net  proceeds, 
agent’s  commission  20  per  cent.? 

1/.  Find  the  gross  proceeds  of  a sale  yielding  $70  net,  charges  $2,  commis- 
sion 10  per  cent. 

5.  A manufacturer  furnishes  bicycles  to  an  agent  who  returns  him  $60 
each  after  taking  out  his  commission  of  16§  per  cent,  and  deducting  $5  charges; 
what  is  the  gross  cost  to  the  purchaser? 

6.  An  agent’s  commission  is  $20  and  his  rate  of  commission  is  5 per  cent. ; 
what  is  the  amount  of  sales  ? 

7.  A dealer  averages  a daily  commission  of  $40  on  a 5 per  cent,  rate ; what 
amount  of  sales  does  he  average  daily  ? 

8.  How  many  sewing  machines  at  $50  each  must  an  agent  sell  to  realize 
a commission  of  $100  weekly  upon  a commission  of  20  per  cent.? 

9.  What  amount  of  books  must  an  agent  sell  to  earn  $200  a month  at  a 
commission  of  40  per  cent.? 

10.  A man  can  earn  $80  a month  on  a 20  per  cent,  commission;  what  rate 
per  cent,  of  commission  must  he  receive  on  the  same  amount  of  sales  to  earn 
$100  a month  ? 

11.  What  sum  was  obtained  by  a collector  who  received  $45  at  a rate  of  5 
per  cent.  ? 

12.  What  amount  of  stocks  was  sold  by  a broker  whose  brokerage  was  $10 
at  a rate  of  \ per  cent.? 

13.  A landlord  received  from  his  agent  a net  monthly  rental  of  $45  after  $3 
was  deducted  for  repairs  and  a commission  of  4 per  cent.;  what  did  the  tenant 
pay  per  month  ? 

11/..  A real  estate  agent  receives  $72  for  the  sale  of  a property  at  5 per  cent, 
commission  ; what  was  the  value  of  the  property? 

15.  What  amount  of  sales  would  a commission  merchant  need  to  make  at 
2J  per  cent,  to  bring  in  $300  commission  a month  ? 

16.  The  gross  proceeds  are  $1275,  the  rate  of  commission  4 per  cent,  and 
the  other  expenses  $18.30.  What  are  the  net  proceeds? 

17.  The  net  proceeds  are  $77.90  and  the  rate  of  commission  5 per  cent. 
What  are  the  gross  proceeds  ? 

18.  How  much  do  I realize  on  a house  and  lot  sold  on  commission  of  3 per 
cent.,  the  total  charges,  including  $25  for  advertising,  being  $130? 


176 


COMMISSION 


WRITTEN  PROBLEMS 


583.  1.  An  agent  retained  $11.25  as  bis  commission  at  2^%  ; what  amount 
did  lie  send  his  principal? 

2.  What  amount  of  goods  must  a salesman  sell  in  a year  at  3%  to  yield 
him  an  income  of  $5000? 

3.  A collector  received  $7500  of  doubtlul  debts  to  collect  and  obtained  75% 
of  the  amount;  if  be  charges  7J%  for  collecting,  what  amount  shall  be  hand  his 
employer? 

A broker  charged  $152.40  at  3|%  for  selling  goods;  what  was  the 
amount  of  sales? 

5.  If  the  brokerage  for  buying  is  $166,  rate  2J%,  what  is  the  value  of  the 
purchase  ? What  is  the  cost  to  the  principal  ? 

6.  How  many  pounds  of  tea  at  42  cents,  did  an  agent  sell  if  he  remitted  his 
principal  $953.30  after  deducting  his  commission  at  3J%  ? 

7.  How  many  dollars’  worth  of  goods  can  an  agent  buy  with  $14000, allowing 
him  a commission  of  3J  % ? 

8.  The  net  proceeds  of  an  account  sales  were  $3060.89  ; what  were  the 
gross  proceeds,  if  the  commission  was  5%,  cartage  $13.60,  storage  $S.30  aud 
insurance  $3  ? 

9.  How  many  barrels  of  flour,  at  $5.122,  can  my  agent  buy  for  me  with 
$6340.96,  if  his  commission  is  2%  ? 

10.  A consigned  to  a commission  merchant  346  barrels  of  potatoes  and  218 
barrels  of  onions  with  instructions  to  sell  and  invest  the  proceeds  in  hay.  The 
commission  merchant  paid  $14  60  freight  and  $6.20  cartage,  and  sold  the  potatoes 
at  $1.40  per  barrel  and  the  onions  at  $3.20  per  barrel,  charging  5%  commission 
for  selling.  He  then  purchased  the  hay  at  $12.25  per  ton,  charging  2%  for  buy- 
ing. How  much  hay  did  A receive? 

584.  To  find  the  rate  per  cent,  of  commission  when  the  commission 
and  the  gross  proceeds  or  the  prime  cost  are  given. 

Example. — Received  from  my  agent  an  account  sales,  showing  my  net  pro- 
ceeds to  be  $2649.89  and  his  commission  $96.11 ; what  rate  did  he  charge  me? 


$2649.89  net  proceeds 


96.11  commission 


035 

2746)9611 
82  38 


Or,  2746)  96.110  (.035 


8238 

13730 

13730 


$2746.00  gross  proceeds 


13730 

13730 


Result,  34%. 

585.  Rule. — Divide  the  commission  by  the  gross  proceeds  or  by  the  prime  cost. 


COMMISSION’ 


177 


MENTAL  PROBLEMS 

586.  1.  What  rate  per  cent,  of  commission  would  I receive  if  I earned  $5 
by  selling  a wheel  for  $50  ? 

Solution. — $5  is  r\,  of  the  selling  price,  which  is  equivalent  to  10  per  cent. 

2.  I am  offered  $3  to  guarantee  a debt  of  $100;  what  is  the  rate  of 
guaranty  ? 

3.  An  agent  returned  $45  on  a sale  of  $50 ; what  rate  of  commission  did  he 
charge  ? 

If.  An  agent  sold  a book  for  $10  and  returned  $6  to  the  publisher;  what 
was  the  agent’s  rate  of  commission  ? 

5.  A broker  returned  $399  for  the  sale  of  a $400  bond  at  its  face  value  ; 
what  was  the  rate  of  brokerage? 

6.  An  agent  gets  $12  for  every  set  of  cyclopedias  he  sells  for  $60;  what  is 
his  rate  of  commission? 

7.  A merchant  offers  a collector  $25  to  collect  $300  worth  of  accounts  ; what 
rate  does  he  offer? 

8.  What  rate  does  a real  estate  agent  receive  for  selling  a property  worth 
$1600  and  retaining  $80  commission  ? 

9.  A drover  is  offered  $20  to  sell. a horse  for  $160;  what  rate  of  commission 
would  he  receive  ? 

10.  A man  sells  wheels  on  commission  and  receives  $10  on  a $60  wheel ; 
what  rate  of  commission  does  he  receive  ? 

11.  What  rate  of  commission  is  charged  by  an  auctioneer  who  receives  $5 
for  selling  a carriage  for  $125? 

12.  A consignor’s  net  proceeds  of  a sale  were  $380  ; the  commission  was  $10 
and  other  charges  $10 ; what  was  the  rate  of  commission? 

13.  What  rate  of  commission  does  an  agent  retain  who  remits  his  principal 
$350  from  a sale  of  $400  ? 

Ilf.  If  an  agent  earns  $30  by  selling  $180  worth  of  goods,  what  is  his  rate  of 
commission  ? 

15.  A salesman  receives  $S  for  selling  $160  worth  of  goods;  what  rate  does 
he  receive  for  selling? 

16.  What  is  the  rate  of  commission  if  the  first  cost  of  merchandise  is  $480 
and  the  commission  $4.20  ? 

17.  If  goods  were  sold  for  $500,  and  only  $490  are  returned  to  the  principal, 
what  per  cent,  is  charged  ? 

18.  On  a purchase  of  $900  worth  of  goods,  $913.50  was  paid  as  the  total 
cost.  What  rate  was  charged  for  buying? 


178 


COMMISSION 


WRITTEN  PROBLEMS 

587.  1.  A lawyer  took  a fee  of  $35  for  collecting  a claim  of  $700;  what  was 
his  rate  for  collecting  ? 

2.  A collector  turned  over  to  his  employer  $2443.59,  retaining  a commission 
of  $128.61  ; wdiat  was  his  rate  for  collecting?  What  was  the  amount  collected? 

3.  If  a merchant  receives  $62.50  for  selling  goods  to  the  amount  of  $2500, 
what  amount  of  commission  should  he  receive  when  he  sells  to  the  amount 
of  $3300  ? 

J.  A collector  receives  $16  as  his  commission  for  collecting  $320,  and,  later 
on,  $28  for  collecting  $560 ; what  is  his  rate  for  collecting  ? 

5.  On  an  estate  of  $32500  an  inheritance  tax  of  $162.50  was  paid  ; what 
was  the  tax  rate  ? 

6.  What  per  cent,  commission  does  an  agent  receive  if  he  renders  an 
account  purchase  showing  total  cost  $1428.91,  his  commission  $34.37,  and  other 
charges  $19.74? 

7.  A commission  merchant  rendered  an  account  sales  showing  $334.08  net 
proceeds  and  his  commission  $13.92.  What  rate  did  he  charge? 

8.  An  agent  received  a consignment  of  goods  which  lie  sold  for  $32414.61, 
charging  1J%  commission.  He  invested  the  proceeds  in  other  goods,  buying 
$31312  worth.  What  per  cent,  did  he  receive  for  buying? 

9.  Jones  consigned  to  a commission  merchant  563  bus.  of  wheat  and  1224 
bus.  of  corn.  The  merchant  sold  the  wheat  at  82§  cents  a bushel  and  the  corn 
at  67f  cents  a bushel.  His  charges  were  $90.72,  including  bis  commission  of 
$51.66.  What  was  the  rate  of  his  commission,  and  what  were  the  net  proceeds? 

10.  I purchase  through  an  agent  in  London  1244  yds.  of  cloth  at  14s.  3d.  a 
yard,  and  remit  to  him  in  payment  a draft  which  costs  me  $4548.10.  at  the  rate 
of  $4.88  for  £1.  What  is  the  rate  of  his  commission,  if  the  other  charges  amount 
to  £26  8s.  6 d? 


GENERAL  PROBLEMS  IN  COMMISSION 

588.  1.  A lawyer  received  a claim  of  $624.50  to  collect.  He  succeeded  in 
collecting  $447.25.  What  per  cent,  did  the  holder  of  the  claim  receive,  if  the 
lawyer’s  commission  for  collecting  was  5%  ? 

2.  What  is  the  commission,  at  2J%,  on  a sale  of  268  bus.  of  oats  at  3S2  cents 
a bushel  ? 

3.  What  will  7200  doz.  eggs  cost  at  18f  cents  a dozen,  commission  3%  ? 

J.  An  agent  remits  $319.60  proceeds  of  a sale  amounting  to  $33S  20  : what 
rate  of  commission  does  he  receive  ? 

5.  A consignor’s  net  proceeds  are  $2314.80,  the  charges  being  $67.12  and 
5%  commission.  Find  amount  of  sale. 

6.  A broker  charges  $72  for  selling  80  bales  of  cotton,  averaging  480  lbs. 
each,  at  7J  cents  a pound.  What  is  the  rate  of  his  commission? 


COMMISSION 


179 


7.  Sold  750  bus.  com  at  56  cents  per  bushel,  and  460  bus.  rye  at  82  cents 
per  bushel.  Commission  3%,  freight  $264,  storage  $86.  Find  net  proceeds. 

8.  How  much  must  I remit  to  my  agent  for  a purchase  of  $378.90  worth  of 
goods,  at  2%  commission? 

9.  Received  $598.55  as  net  proceeds  of  a consignment  on  which  the  com- 
mission amounted  to  $20.37  and  the  expenses  $60.  Find  rate  of  commission. 

10.  A real  estate  broker  charged  me  $382.50  as  his  commission  at  5%  for 
selling  a house.  How  much  did  I receive? 

11.  Remitted  my  agent  $690.45  to  invest  in  cotton  at  8|  cents  per  pound  ; 
how  many  pounds  did  he  buy,  his  commission  being  2J%  ? 

12.  What  are  the  net  proceeds  of  a consignment  of  174  barrels  of  potatoes, 
of  which  67  barrels  are  sold  at  $3.50  per  barrel,  24  barrels  at  $3.40  per  barrel,  and 
83  barrels  at  $3.20  per  barrel,  the  charges  being  $37.40,  and  commission  3%  ? 

13.  What  is  the  rate  of  commission  on  a consignment  which  yields  $339.68 
net  proceeds,  the  commission  being  $12.62  and  the  other  charges  $26.30? 

74-  A commission  merchant  remitted  $408.98  as  net  proceeds  of  a consign- 
ment on  which  the  charges  were  $8.90  and  his  commission  24%.  How  much 
was  his  commission? 

15.  If  I remit  to  my  agent  $4000  to  invest  in  flour,  how  many  barrels  can 
he  buy  at  $4.90,  and  what  is  the  balance  left  over,  if  his  commission  is  2f%  ? 

16.  What  is  a stock  broker’s  commission,  at  §%,  for  purchasing  $10000 
worth  of  bonds  ? 

17.  Find  the  net  proceeds  of  the  following  : 


Account  Sales 

Philadelphia,  July  15,  1908. 


Sold  for  account  of  E.  N.  Welsh, 


By  J.  T.  Rodman  & Co. 


350  bbls.  Greening  Apples 
175  “ Russet  “ 

412  “ Baldwin  “ 

Charges. 

Freight 

Cartage 

Commission  and  guaranty,  5% 
Net  proceeds 


@ $3.70 
3.85 
3.65 


595 

58 

60 

50 

18.  Machines  that  cost  $123.25  are  listed  at  $200.  If  a discount  of  12|%, 
10%  and  5%  is  allowed,  and  a commission  of  10%  paid  the  salesman,  what  per 
cent,  is  gained  ? 


180 


COMMISSION 


19.  An  attorney  collected  part  of  a claim,  charging  $12.30  for  his  commis- 
sion of  5%.  If  the  part  collected  was  but  35%  of  the  entire  claim,  what  was  the 
amount  of  the  loss  ? 

50.  A Minneapolis  merchant  shipped  to  his  agent  in  New  York  700  barrels 
of  wheat  flour  and  2000  barrels  of  rye  flour.  The  agent  sold  the  wheat  flour  at 
$3.50  per  barrel  and  the  rye  flour  at  $1.75  per  barrel.  After  taking  out  his  com- 
mission he  remitted  $5801.25.  Find  per  cent,  of  commission. 

51.  An  agent  receives  $7500  to  invest  in  wheat  at  72f  cents  a bushel,  allow- 
ing 3%  commission.  What  amount  of  wheat  can  he  buy  ? 

SS.  An  importer  buys  through  his  agent  in  Paris  1217  meters  of  silk  at 
17.28  francs  per  meter,  paying  the  agent  5%  commission  and  expenses  213 
francs.  If  the  duty  is  $1.70  per  yard,  at  what  price  per  yard  must  the  silk  he 
sold  in  Philadelphia  to  gain  20%  ? 

S3.  A broker  charged  $17  as  his  commission  at  \ % for  buying  U.  S.  govern- 
ment bonds.  What  amount  of  bonds  did  he  purchase? 

Slf..  I send  an  agent  $2472.80  to  invest  in  cotton  at  $5.23  per  bale,  allowing 
5%  commission.  He  pays  14  cents  a bale  freight,  2 cents  a bale  storage  and  34 
cents  a bale  drayage.  How  many  hales  does  he  purchase,  and  what  is  the 
unexpended  balance? 

55.  An  agent  sells  a consignment  of  12000  bushels  of  wheat  at  70^  cents  a 
bushel,  deducts  his  commission  of  2%,  and  invests  the  net  proceeds  in  flour  at 
$3.72  per  barrel,  charging  2%  commission  for  buying.  How  many  barrels  does 
he  buy,  and  what  is  the  balance  unspent  ? 

56.  Find  the  net  proceeds  of  the  following: 

Account  Sales 


Sold  for  account  of  Brown  & Sharp, 


Philadelphia,  May  4.  1908. 
By  Marshall  Brothers. 


1908 

Jan. 

29 

387  yds  Broadcloth 

@ $4.63 

Feb. 

13 

249  “ 

3.78 

Mar. 

10 

426  “ 

6.20 

Apr. 

29 

512  “ 

5.50 

Charges. 

§27-12  $6.50 

Freight  and  Drayage 

33  62 

Commission,  5% 

i 

Net  proceeds 

INSURANCE 


589.  Insurance  is  “ a contract  by  which  one  party,  for  an  agreed  consider- 
ation (which  is  proportioned  to  the  risk  involved),  undertakes  to  compensate  the 
other  for  loss  on  a specified  thing  from  specified  causes.  The  party  agreeing  to 
make  the  compensation  is  usually  called  the  insurer  or  underm' iter  ; the  other,  the 
insured  or  assured  ; the  agreed  consideration,  the  premium  ; the  written  contract,  a 
policy ; the  events  insured  against,  risks  or  perils;  and  the  subject,  right,  or 
interest  to  be  protected,  the  insurable  interest — Bouvier. 

590.  The  principal  kinds  of  insurance  are  fire  insurance,  marine  insurance, 
accident  insurance,  and  life  insurance. 

591.  Fire  insurance  provides  indemnity  for  loss  of  or  damage  to  property 
by  fire  or  the  efforts  to  extinguish  fire. 

592.  Marine  insurance  provides  indemnity  for  loss  by  shipwreck  or 
disaster  at  sea. 

593.  Accident  insurance  provides  indemnity  for  loss  occasioned  by  explo- 
sion, breakage,  or  other  accidents  to  property,  or  loss  of  future  earnings  through 
personal  disablement. 

594.  Life  insurance  is  a contract  by  which  a company,  in  consideration 
of  certain  premiums,  agrees  to  pay  to  the  heirs  of  a person  when  he  dies,  or  to 
himself  if  living  at  a specified  time,  a certain  sum  of  money. 

595.  Adjustment  of  losses.  Fire  insurance  companies  pay  the  full 
amount  of  the  loss,  up  to  the  limit  of  the  policy,  unless  the  contract  contains  the 
“ average  clause,”  which  provides  that  the  company  shall  pay  only  such  propor- 
tion of  the  loss  as  the  sum  insured  bears  to  the  full  value  of  the  property.  Marine 
insurance  policies  usually  contain  the  ‘‘  average  clause”. 

Note. — Under  the  “ average  clause,  ” if  property  worth  $8000  is  insured  for  f of  its  value,  or 
$6000,  the  company,  in  case  of  loss,  will  pay  f of  the  loss;  that  is,  in  case  of  a loss  of  $4000,  the  com- 
pany would  pay  $3000;  in  case  of  a loss  of  $3000,  it  would  pay  $2250,  etc.  If  the  property  were 
insured  for  \ of  its  value,  the  company  would  pay  but  I of  the  loss,  etc. 

596.  To  find  the  premium. 

Example. — What  is  the  premium  on  $12000  insurance 
atlj%? 

The  other  cases  under  insurance  are  more  theoretical  than 
practical,  and  are  simply  applications  of  percentage  principles 
already  explained. 

597.  Rule. — Multiply  amount  of  insurance  named  in  policy  by  the  rate  per 
cent,  of  premium. 


//  2 0 O 0 
.0  / 
/ 2 O O 

fo.o  0 


181 


182 


INSURANCE 


MENTAL  PROBLEMS 

598.  1.  What  is  the  premium  on  an  insurance  policy  of  $2000  at  2 per 
cent.? 

Solution. — The  premiun  is  T§ 7 or  fa  of  $2000,  which  is  $40. 

2.  What  is  the  premium  on  a policy  of  $1200  at  2J  per  cent,? 

3.  What  is  the  premium  on  a policy  of  $5000  at  1J  per  cent.? 

I/..  What  premium  must  I pay  for  insuring  a vessel  worth  $8000  for  f-  of  its 
value,  at  2 per  cent.? 

5.  My  store,  worth  $3600,  is  insured  for  of  its  value  at  1 J per  cent,  and  its 
contents,  valued  at  $8000,  for  f of  their  value  at  2 per  cent ; what  is  the  entire 
premium  ? 

6.  My  charge  for  insuring  a house  worth  $6000  for  § of  its  value  is  11  per 
cent.,  what  is  the  amount  of  my  bill  ? 

7.  My  property,  worth  $9000,  is  insured  for  J of  its  value  in  a company  at 
1J  percent,  and  for  J of  its  value  in  a company  at  2 per  cent.;  what  amount  of 
insurance  do  I carry,  and  what  does  it  cost  me? 

8.  A merchant  has  a stock  valued  at  $12000  which  one  company  offers  to 
insure  for  f of  its  value  at  11  per  cent.,  and  another  for  i of  its  value  at  1 J per 
cent.;  what  will  be  the  difference  in  cost  between  the  premiums? 

9.  Which  is  better  and  how  much — an  insurance  on  $1200  at  21  per  cent.  • 
for  three  years,  or  $1200  at  1 per  cent,  per  annum  ? 

10.  What  is  the  amount  of  premium  on  furniture  worth  $4200  at  11  per 
cent.  ? 

11.  At  2 per  cent,  what  will  be  the  face  of  a policy  whose  premium  is  $30? 

Solution. — 2 per  cent,  or  fa  of  policy  equals  $30,  hence  face  of  policy  is  50  times  $30,  or  $1500. 

12.  At  50  cents  for  $100,  how  much  insurance  can  be  bought  for  $20  ? 

13.  If  the  rate  of  insurance  is  2J  per  cent,  and  premium  $50,  what  is  the 
face  of  the  policy  ? 

Ilf.  An  insurance  policy  cost  $45,  premium  at  21  per  cent.  What  was 
amount  named  in  policy  ? 

15.  If  I pay  $50  premium  quarterly  on  insurance  at  11  per  cent.,  how  much 
insurance  do  I carry? 

16.  What  is  the  face  of  a policy  whose  premium  is  $30  a quarter,  at  11  per 
cent.? 

17.  $23  is  charged  me  upon  my  obtaining  an  insurance  policy  at  f per  cent, 
premium  ; what  is  the  face  of  my  policy? 

18.  If  an  agent’s  commissions  for  writing  insurance  during  a week  amount 
to  $100  at  1 per  cent,  commission,  what  amount  has  he  written  ? 

19.  If  a company  has  received  premiums  during  a week  amounting  to  $200, 
written  at  I per  cent.,  what  amount  of  insurance  has  it  issued  ? 

20.  I write  insurance  for  a company  which  charges  2 per  cent,  and  allows 
me  1 per  cent.  If  I earn  $60  a week,  what  amount  must  the  company  issue  ? 


INSURANCE 


183 


WRITTEN  PROBLEMS  IN  INSURANCE 

599.  1 ■ What  is  the  premium  at  If  % on  $6250  insurance? 

2.  If  $105  is  paid  as  premium  for  five  years  on  a policy  of  $3500,  what  is 
the  rate  per  annum  ? 

3.  At  the  rate  of  2f%,  $264.69  was  paid  as  premium;  what  was  the 
amount  of  the  risk  ? 

4-  What  is  the  cost  of  insuring  a cargo  valued  at  $5674.50  for  75%  of  its 
value,  at  the  rate  of  If  % ? 

5.  If  the  cargo  mentioned  in  the  preceding  problem  were  damaged  to  the 
extent  of  $820,  how  much  would  the  policy  holder  receive  from  the  insurance 
company  ? 

6.  A insured  his  house  on  March  1,  1906,  for  $7500  for  5 years,  at  3J%. 
What  was  the  amount  of  “unearned  premium”  on  Sept.  1,  1908? 

7.  A cargo  worth  $22587.36  was  insured  for  80%  of  its  value  at  lf% 
premium.  The  ship  was  lost  at  sea.  How  much  did  the  insurance  company  lose  ? 

8.  A merchant  insured  his  stock  of  goods  with  one  company  for  $5000, 
with  another  for  $3000,  and  with  a third  for  $2500.  If  the  goods  should  be  dam- 
aged by  fire  to  the  extent  of  $4000,  how  much  should  each  company  pay  ? 

9.  What  is  an  insurance  broker’s  commission  at  20%,  for  placing  $40000 
insurance  at  f % ? 

10.  What  per  cent,  of  loss  would  a marine  insurance  company  pay  on  goods 
valued  at  $4678.18  and  insured  at  $3500? 

11.  A man  had  his  life  insured  for  $5000,  paying  an  annual  premium  of 
$162.85.  After  paying  premiums  for  12  years  he  died.  What  per  cent,  more 
did  the  company  pay  the  beneficiary  than  had  been  received  in  premiums? 

12.  B insured  his  house  in  one  company  at  If  % premium,  and  in  another 
at  f%  premium.  His  total  premium  was  $105.  If  the  amount  of  the  second 
policy  was  half  that  of  the  first,  what  was  the  amount  of  each? 

13.  A cargo  was  insured  for  equal  amounts  in  three  different  companies,  at 
a premium  of  If  % in  the  first,  1%  in  the  second,  and  lf%  in  the  third,  the 
total  insurance  being  f of  the  value  of  the  cargo.  The  cargo  was  lost  by  ship- 
wreck, and  the  owner  thereby  lost  $7135  (including  the  premiums  he  had  paid). 
For  what  amount  were  the  policies  drawn  ? 

14.  What  is  merchandise  worth  that  is  insured  for  f-  of  its  value  at  a 
premium  of  ^0%,  if  the  premium  amounts  to  $57.04? 

15.  A building  and  its  contents  are  separately  insured,  the  building  for  f of 
its  value  at  f % premium,  and  the  contents  for  -4  of  their  value  at  f-%  premium. 
The  premium  for  insuring  both  is  $1091.67.  If  the  value  of  the  contents  is 
$85000,  how  much  is  the  building  worth? 

16.  For  how  much  must  goods  worth  $25000  be  insured,  at  f % premium,  in 
order  that  the  owner  may  lose  nothing  if  they  are  totally  destroyed  by  fire? 


184 


INSURANCE 


17.  Certain  insurance  policies  contain  the  following  co-insurance  clause: 

If  at  the  time  of  fire  the  whole  amount  of  insurance  on  the  property  covered  by  each  separate 
item  of  this  policy  on  property  as  described  in  such  item  shall  be  less  than  80%  of  the  actual  cash  value 
thereof,  this  company  shall  in  case  of  loss  or  damage  be  liable  for  only  such  proportion  of  such  loss  or 
damage  as  the  amount  insured  under  said  item  shall  bear  to  the  said  80%  of  the  actual  cash  value  of 
the  property  covered  by  such  item.” 

Under  the  provisions  of  this  clause,  how  much  would  each  company  pay  in 
case  of  damage  amounting  to  $420  on  property  worth  $12000,  insured  for  $3000 
in  each  of  three  companies?  How  much  would  each  pay  if  the  property  had 
been  insured  for  $3500  in  each  company? 

18.  Under  the  80%  co-insurance  clause,  as  above,  how  much  would  be 
recovered,  in  case  of  total  loss,  on  property  valued  at  $60000  and  insured  to  the 
extent  of  $48000?  How  much  on  property  valued  at  $60000  and  insured  for 
$40000  ? 

19.  A factory  (worth  $3000)  and  its  contents  were  insured  for  $10000,  as 
follows:  $2000  on  building,  $3000  on  machinery  (worth  $5000),  and  $5000  on 
stock  (worth  $8000).  The  building  was  damaged  by  fire  to  the  amount  of  $1000, 
the  machinery  $4000,  and  the  stock  was  a total  loss.  How  much  was  the  claim 
against  the  insurance  company?  What  was  the  premium  at  1J%  ? What  was 
the  owner’s  loss?  What  was  the  company’s  loss,  if  the  risk  was  covered  (1)  by 
an  “ordinary  policy;”  (2)  if  the  policy  contained  the  “average  clause;”  (3)  if 
the  policy  contained  the  “co-insurance  clause?” 


REVIEW  PROBLEMS  IN  PERCENTAGE 

Remark. — It  will  be  noticed  that  no  attempt  has  been  made  to  construct  problems  that  would 
produce  certain  cut-and-dried  results.  Problems  are  presented  just  as  they  arise  in  business  and  the 
same  kinds  of  results  are  obtained  as  would  be  obtained  in  actual  business.  While  it  would  be  a 
curious  thing  to  find  that  it  required  ff  of  a man  to  plow  a certain  field  of  a certain  extent  with  a team 
walking  a certain  gait,  still  such  fractional  part  of  a man  is  a perfectly  logical  and  reasonable  mathe- 
matical fact. 

In  rates  per  cent.,  which  come  out  so  nicely  in  arithmetics,  it  is  possible,  and,  in  truth,  more 
than  likely,  that  a very  unusual  fraction  may  appear.  Where  an  approximate  result  will  answer,  we 
suggest  that  the  rate  be  carried  to  two  places  beyond  the  usual  number  and  the  rate  be  expressed  as  an 
integer  and  decimal.  Thus,  say  $4.78  is  1.59%  of  $299  ; or  the  decimal  part  may  be  reduced  to  its 
approximate  common  fraction,  which  in  this  case  is  or  f,  so  that  If  % very  nearly  expresses  the 
rate.  Wherever  exactness  is  required,  however,  as  in  distributing  moneys  among  a large  number  of 
persons,  it  may  be  necessary  to  carry  the  rate  out  to  four  or  more  decimal  places. 

600.  1.  A railway  company’s  earnings  for  the  years  1907  and  1908  were 
$3648729.13.  The  earnings  for  1908  were  14%  more  than  for  1907.  Fmd 
earnings  for  each  year. 


REVIEW  PROBLEMS 


185 


2.  What  per  cent,  above  cost  must  goods  be  marked  to  allow  a discount  of 
20  % and  5 % and  still  yield  a profit  of  16§  % ? 

3.  One-fourth  of  a consignment  of  goods  is  sold  at  a loss  of  18%,  and  one- 
third  at  a gain  of  6J%.  For  how  much  above  cost  must  the  balance  be  sold  to 
net  a profit  of  20%  on  the  whole  consignment  ? 

If..  In  a certain  compound  the  weights  of  the  ingredients  are,  respectively, 
4 oz. ; 2 oz.  13  pwt. ; 20  gr. ; and  1 oz.  4 gr.  Find  the  per  cent,  of  each 
ingredient  ? 

5.  Sold  a bicycle  for  10%  less  than  it  cost  me.  If  I had  sold  it  for  $60,  my 
gain  would  have  been  20%.  How  much  did  I lose? 

6.  If  A sells  goods  at  list  priceless  40%,  25%  and  10%,  and  B sells  at  same 
list  less  30%,  10%  and  10%,  what  per  cent,  of  A’s  net  selling  price  is  B’s  ? 

7.  How  much  is  Jones’s  investment  if  his  income,  at  8%,  is  $1893.76? 

8.  Consigned  merchandise  to  an  agent,  which  he  sold  for  $3211.22,  charging 
2 \f0  commission.  According  to  instructions,  he  invested  the  proceeds  in 
wheat  after  taking  out  his  commission  of  2%  for  purchasing.  How  much  did  he 
have  to  invest  ? 

9.  A sold  B a horse  for  $114,  which  is  5%  less  than  A paid  for  him.  B 
sells  the  horse  for  15%  more  than  it  cost  A.  How  much  did  B receive  for  the 
horse? 

10.  What  will  be  the  cost  of  insuring  a house  for  $7500  at  f %,  and  the 
furniture,  etc.,  for  $3000  at  f % ? 

11.  After  retaining  3J%  commission  for  selling  a consignment  of  cotton, 
my  agent  paid  me  $4174.59.  How  much  was  his  commission  ? 

12.  Sold  a farm  for  $3904.32,  thereby  losing  17%.  The  sum  I paid  for  the 
farm  was  what  I received  as  my  one-third  share  of  a legacy,  after  the  executor 
had  received  2%  for  settling  the  estate.  What  was  the  value  of  the  entire  estate? 

13.  A property  being  sold  at  23|%  above  cost,  the  gain  amounted  to 
$1741.92  ; what  was  the  cost  ? 

Ilf.  In  a school  examination,  James  obtained  the  correct  results  to  7 prob- 
lems out  of  10  in  arithmetic,  spelled  correctly  34  out  of  40  words,  answered 
correctly  14  out  of  15  questions  in  geography,  19  out  of  23  in  grammar,  and  13 
out  of  18  in  history;  what  per  cent,  general  average  did  he  make? 

15.  If  the  above  examination  had  been  averaged  according  to  the  U.  S. 
civil  service  method,  and  arithmetic  had  been  given  a value  of  6,  spelling  5, 
geography  3,  grammar  4,  and  history  2,  what  would  have  been  his  general 
average  ? 

16.  What  per  cent,  profit  is  made  by  buying  wine  at  2 francs  a liter  and 
selling  it  at  $2  a gallon  ? 

17.  Bought  goods  for  $382.75 ; what  price  must  I ask  for  them  so  as  to  be 
able  to  allow  a discount  of  25%,  10%  and  5%,  and  net  a profit  of  20%  ? 

18.  If  the  par  value  of  the  stock  of  a certain  railway  is  $50  per  share,  and 
it  is  selling  for  97f  % of  its  par  value,  what  will  150  shares  cost  if  purchased 
through  a broker  who  receives  $9.38  commission  ? 


186 


REVIEW  PROBLEMS 


19.  On  June  8,  1908,  E.  S.  Grayson  & Bro.,  commission  merchants,  of  Phila- 

delphia, received  from  L.  N.  Clark,  of  Chester,  Pa.,  a consignment  of  1437  lbs.  of 
butter,  to  be  sold  for  his  account  and  risk.  They  paid  $6.70  charges,  and  sold 
the  butter  as  follows:  On  June  10,  329  lbs.  @ 35 cents;  June  12, 412  lbs.  @ 41 J 

cents ; June  14,  224  lbs.  @ 37|  cents  ; June  15,  118  lbs.  @ 36J  cents  ; June  17, 
272  lbs.  @ 39  cents;  June  18,  82  lbs.  @ 38J  cents.  On  June  21  they  rendered  an 
account  sales,  charging  5%  commission.  Make  out  the  account  sales  in  proper 
form,  showing  Clark’s  net  proceeds. 

20.  A man  having  a certain  sum  of  mone}r,  invested  J of  it  in  wheat,  J of  it 
in  oil,  and  the  remainder  in  cotton.  He  sold  the  wheat  at  a profit  of  6%,  the  oil 
at  a loss  of  2%,  and  the  cotton  at  a profit  of  9%.  If  the  total  amount  received 
from  the  sales  was  $6230,  how  much  had  he  at  first? 

HI.  If  a grocer  pays  $57.60  for  180  lbs.  of  Mocha  coffee,  how  many  pounds 
of  Java  coffee  costing  26  cents  per  pound  must  he  mix  with  it  in  order  to  gain 
20%  by  selling  the  mixture  at  36  cents  per  pound? 

22.  What  is  6%  of  £43  13s.  8 d.  ? 

23.  A sold  goods  to  B for  $967.20  and  gained  3%.  B sold  them  to  C and 
lost  3%.  How  much  more  or  less  did  C pay  than  A ? 

24-.  What  per  cent,  larger  than  a gallon  is  a half  peck  ? 

25.  If  I lose  10%  by  selling  cloth  at  72  cents  per  yard,  at  what  price  must  I 
sell  it  to  gain  25  % ? 

26.  The  net  amount  of  a bill  of  goods  is  $3783.18,  the  discounts  allowed  being 
35%,  15%  and  10%,  and  2%  for  cash.  What  is  the  gross  amount  of  the  bill? 

27.  The  capital  stock  of  a railroad  is  $28500000.  Its  gross  earnings  for  the 
year  1907  were  $3642819.76,  and  its  operating  expenses  $1938743.29.  After 
paying  $825000  interest  on  its  bonded  debt,  what  per  cent,  of  its  earnings  remains  ? 

28.  If  an  agent’s  commission  for  selling  $3264  worth  of  goods  is  $96.92,  how 
much  would  he  receive,  at  the  same  rate,  for  selling  $7492.85  ? 

29.  Bought  two  houses,  28%  of  the  cost  of  the  first  being  equal  to  20%  of 
the  cost  of  the  second.  If  the  second  cost  $2000  more  than  the  first,  what  was  the 
cost  of  each  ? 

30.  By  selling  40%  of  a purchase  at  20%  profit,  65%  of  the  remainder  at 
15%  profit,  and  what  then  remained  at  10%  loss,  a net  gain  of  $468.69  was  made. 
Find  amount  of  purchase. 

31.  A,  B and  C are  partners.  30%  of  A’s  share  in  the  business  equals  18% 
of  B’s  share,  and  12%  of  B’s  share  equals  40%  of  C’s  share.  C’s  share  is  $5400. 
Find  A’s  and  B’s  shares. 

32.  Robertson  withdrew  his  money  from  a business  that  was  yielding  him 
an  annual  income  of  8g-%,  and  invested  in  another  business  that  turned  out  less 
profitable  than  he  expected,  yielding  only  74%  on  his  investment.  His  annual 
income  was  reduced  $106.25  by  the  change.  How  much  capital  had  he  invested? 


REVIEW  PROBLEMS 


187 


33.  An  attorney  succeeded  in  collecting  70%  of  a claim  placed  in  his  hands, 
and  received  $55.26  as  his  commission  at  2%.  What  was  the  whole  amount  of 
the  claim  ? 

34-.  Half  of  a cargo  that  cost  $32468.18  was  sold  at  12%  profit,  and  20%  of 
it  at  a loss  of  7|%.  For  how  much  must  the  remainder  be  sold  to  yield  a net 
gain  of  10  % on  the  whole  ? 

35.  If  a dealer  sells  f of  a pound  for  what  of  a pound  cost,  what  per  cent, 
does  he  gain  ? 

36.  What  will  it  cost  me  in  U.  S.  money  to  buy,  through  my  agent  at  Brus- 
sels, 580  meters  of  lace  at  6.15  francs  per  meter,  if  the  agent’s  commission  is  3%  ? 

37.  What  per  cent,  profit  would  I make  on  the  lace  in  the  above  problem  if, 
after  paying  freight  and  other  expenses  amounting  to  $89.20,  and  $290  duty,  I 
sell  the  lace  at  $3  per  yard  ? 

38.  What  per  cent,  of  10  francs  is  | of  a pound  sterling? 

39.  How  must  I mark  goods  that  cost  $4.17  a yard,  so  as  to  be  able  to  allow 
a discount  of  20%  and  5%  and  still  gain  25%  ? 

40.  A real  estate  broker  sold  two  houses  for  $3200  and  $4500  respectively. 
The  first  house  brought  23J%  more  than  cost,  and  the  second  10%  less  than  cost. 
If  the  broker’s  commission  for  selling  was  3%,  how  much  did  his  principal  gain 
or  lose  on  both  houses  ? 

41.  My  agent  in  Pittsburg  purchased  for  me  1200  bbls.  of  oil  at  $3  per  barrel. 
He  sold  500  bbls.  of  it  at  $3.75  per  barrel,  and  the  remainder  at  $2.90.  His  com- 
mission was  2%  for  buying  and  3%  for  selling.  How  much  should  he  remit  in 
settlement  of  my  account  ? 

4S.  X bought  a house  for  33£%  more  than  its  assessed  value.  He  made 
improvements  costing  $680,  and  then  sold  the  house  at  a net  gain  of  20%, 
receiving  $11280  for  it.  Find  how  much  the  taxes  on  the  house  amount  to,  at 
$1.80  on  $100. 

43.  After  paying  an  agent’s  commission  of  24%,  $3247.80  were  received  as 
proceeds  of  a sale.  For  how  much  were  the  goods  sold  ? 

44 • H goods  are  bought  at  a discount  of  25%,  10%  and  5%  off  list,  and  sold 
at  same  list  price,  less  15%  and  10%,  what  is  the  gain  per  cent.? 

45.  A grocer  bought  five  dozen  eggs  at  the  rate  of  3 for  5 cents,  and  five 
dozen  at  the  rate  of  2 for  5 cents.  He  sold  them  at  the  rate  of  5 for  10  cents 
under  the  impression  that  he  was  selling  them  at  cost;  what  per  cent,  did 
he  lose? 

46.  What  percent,  of  £8  17  s.  5 d.  is  80.75  francs,  if  £1=25.25  francs? 

47.  From  a cask  containing  217.50  liters  of  wine,  5%  was  lost  by  leakage. 
How  many  quarts  remained? 

48.  How  much  will  it  cost  to  excavate  a cellar  40  ft.  X60  ft.  and  8 ft.  deep, 
35%of  it  being  rock  at  94  cents  a cubic  yard,  and  the  remainder  clay  at  44  cents 
a cubic  yard  ? 


INTEREST 

601.  Interest  is  a charge  made  for  the  use  of  money. 

602.  The  principal  is  the  sum  for  the  use  of  which  interest  is  charged. 

603.  The  rate  is  the  per  cent,  of  the  principal  charged  for  its  use  for  one 
year. 

604.  The  amount  is  the  sum  of  the  principal  and  interest. 

605.  Simple  interest  is  interest  on  the  principal  only. 

606.  Compound  interest  is  interest  on  the  sum  of  the  principal  and  unpaid 
interest ; it  is  interest  on  interest. 

607.  Common  interest  is  interest  computed  on  the  basis  of  12  months 
of  30  days  each,  or  360  days  to  the  year. 

608.  Accurate  or  exact  interest  is  interest  computed  on  the  basis  of  365 
days  to  a year. 

609.  Legal  interest  is  interest  at  the  rate  per  cent,  established  by  law  for 
cases  in  which  the  rate  is  not  specified. 

610.  Accrued  interest  is  the  unpaid  interest  on  an  obligation,  usually  not 
yet  due. 

611.  Usury  is  the  crime  of  charging  a higher  rate  of  interest  than  is 
allowed  by  law. 

612.  To  find  the  interest  on  any  principal. 


GENERAL  FORMULAS 


For  one  year.  Principal  X Rate  (expressed  decimally)  = Interest. 

For  part  of  a year. 

Principal  X Rate  (expressed  decimally)  x interest. 

Example  1. — Find  the  interest  of  $360  for  3 mo.  13  da.  at  6%  per  annum. 


Explanation. — The  interest 
for  1 year  is  .06  times  the  principal, 
or  $21.60.  3 mos.  are  equal  to  90 

days  ; adding  the  13  days  makes  103 
days,  or  Iff  of  a year.  The  interest 
for  of  a year  is  ||f  of  $21.60,  or 
$6.18. 

Or,  by  cancelation  : 


30OX.O6X1O3 


3 


Z 3 & 0 
, O //? 


2 / & O * 

/ 03 


2 / 


FO 

o 


6 0)2  2 2 U.F  O 
' 2 / £>  0 
6 u r 


. / F t /03  eids 


3 4,  O 

2 rro 
2 F F O 


188 


INTEREST 


189 


Example  2. — What  is  the  interest  of  $360  for  2 yr.  3 mo.  13  da.  at  6%  per 
annum  ? 


$360  principal  $21.60X2 

.06  rate  1.80X3 

12)$21.60  interest  1 yr.  .06X13 

30)  1.80  int.  1 mo. 

.06  int.  1 da. 


$43.20  int.  2 yr. 

5.40  int.  3 mo. 

.78  int.  13  da. 

$49.38  int.  2 yr.  3 mo.  13  da. 


Explanation. — The  interest  for  one  year  is  .06  of  the  principal,  or  $21.60;  the  interest  for  one 
month  is  T’5  of  a year’s  interest,  or  $1.80;  the  interest  for  one  day  is  of  a month’s  interest,  or  $.06. 
The  interest  for  two  years  is  twice  one  year’s  interest  or  $43.20;  for  3 mos.  it  is  three  times  one  month’s 
interest,  or  $5.40;  for  13  das.  it  is  thirteen  times  one  day’s  interest,  or  $.78.  Adding,  we  have  $49.38, 
the  interest  for  2 yr.  3 mo.  13  da. 

Remark. — There  is  no  one  application  of  the  general  interest  formulas  that  is  best  for  all  cases. 
The  student  who  has  mastered  the  general  formulas  should  be  led  to  see  clearly  the  special  applications 
and  to  discriminate  in  their  uses.  Let  the  instructor  receive  from  the  student  his  opinion  as  to  which 
particular  application  best  solves  any  particular  problem  under  discussion.  The  business  man  should 
be  skilful  in  using  several  applications  and  use  the  best  method  for  each  case. 

613.  Rule. — To  find  one  year’s  interest,  multiply  the  principal  by  the  rate 
expressed  decimally. 

To  find  the  interest  for  part  of  a year , multiply  one  year’s  interest  by  as  many 
360ths  as  there  are  days.  (Or,  for  each  month,  take  of  a year's  interest ; and  for 
each  day,  of  a month’s  interest. 


MENTAL  PROBLEMS 

614.  1.  I lend  Jones  $100  for  one  year,  interest  at  6 per  cent. ; what  sum 
should  he  pay  me  for  the  use  of  the  monej’  ? 

Solution. — At  6 per  cent,  for  1 year,  the  interest  would  equal  .06  of  the  principal,  which  is  $6; 
hence  he  should  pay  me  $6  interest. 

<2.  What  should  I receive  for  lending  Borrower  $200  for  6 months,  at  6 per 
cent.  ? 

3.  What  should  I return  to  Lender,  principal  and  interest,  for  the  use  of 
$200  for  3 months  at  6 per  cent.? 

f What  would  be  the  yearly  interest  on  a mortgage  of  $400  at  5 per  cent.? 

5.  The  holder  of  a note  for  $400,  bearing  interest  at  6 per  cent.,  receives  the 
interest  in  quarterly  payments.  What  sum  does  he  receive  at  each  payment? 

6.  For  what  amount  should  I draw  a check  in  full  payment  of  a loan  of 
$200  obtained  90  days  ago,  interest  at  6 per  cent.? 

7.  Shaner  lends  me  $300  less  the  interest  on  my  6 months  note,  interest  at 
6 per  cent.;  what  sum  does  he  lend  me? 

8.  Find  the  interest  on  $360  for  4 months  at  4 per  cent. 

9.  What  amount  of  income  shall  I receive  semiannually  on  a $500  bond 
bearing  4 per  cent,  interest? 

10.  What  amount  must  I return  in  6 months  for  a loan  of  $350  at  4 per  cent.? 


190 


INTEREST 


ORAL  EXERCISE 

615.  1.  What  is  the  interest  on  $100  for  1 yr.  at  6%  ? 

2.  On  $200  for  1 yr.  at  6%  ? 

3.  On  $100  for  2 yrs.  at  6%  ? 

J.  On  $200  for  2 yrs.  at  6 % ? 

5.  On  $100  for  2 yrs.  at  5%  ? 

6.  On  $200  for  2 yrs.  at  5 % ? 

7.  On  $200  for  2 yrs.  at  4%  ? 

8.  On  $200  for  1 yr.  6 mo.  (If  yrs.)  at  6%  ? 

9.  On  $300  for  1 yr.  3 mo.  at  4%  ? 

10.  On  $100  for  6 mos.  at  6 % ? 

11.  On  $200  for  3 mos.  at  4%  ? 

I®.  On  $100  for  60  das.  at  6%  ? 

13.  On  $200  for  60  das.  at  5 % ? 

7J.  On  $300  for  90  das.  at  6%  ? 

15.  On  $100  for  120  das.  at  6%  ? 

16.  What  is  the  interest  of  $200  for  2 yrs.  at  5%  ? 

17.  Of  $150  for  1 yr.  6 mo.  at  4%  ? 

18.  Of  $50  for  2 yr.  4 mo.  at  6 % ? 

19.  Of  $3600  for  20  das.  at  2%  ? 

20.  Of  $1000  for  36  das.  at  8%  ? 

WRITTEN  EXERCISE 

616.  1.  Find  the  interest  on  $1530  for  2 yr.  6 mo.  at  6%. 

2.  Find  the  interest  on  $890  for  3 yr.  3 mo.  at  6%. 

3.  What  is  the  amount  of  $364  for  2 yr.  9 mo.  at  6%  ? 

Note. — Principal  + Interest  = Amount. 

J.  What  is  the  interest  on  $1000  from  Jan.  1,  1908,  to  April  1,  1909? 

Note. — When  the  time  extends  beyond  a year,  find  time  by  compound  subtraction. 

5.  Find  the  amount  of  $2520  from  May  1,  1908,  to  November  1,  1909, 
interest  at  5 % . 

6.  Find  the  interest  of  $378  for  1 yr.  4 mo.  at  6%. 

7.  Of  $756  for  2 yr.  2 mo.  15  da.  at  4%. 

8.  Of  $2425.60  for  3 yr.  5 mo.  18  da.  at  5%. 

9.  Of  $589.70  for  84  days  at  7 %. 

10.  Of  $642.50  for  7 mo.  19  da.  at  3%. 

11.  Find  the  amount  of  $342.42  from  Feb.  5, 1907,  to  March  15, 1908,  at  7 

12.  I borrowed  $360.50  Aug.  1, 1907,  and  returned  it  March  5,  1908,  with 
interest  at  6J%.  How  much  was  paid? 

13.  If  $150  was  loaned  Aug.  5,  1906,  and  returned  with  interest  at  7 % on 
Mar.  17,  1908,  how  much  was  paid? 


INTEREST 


191 


WRITTEN  PROBLEMS 

617.  1 ■ What  amount  will  take  up  January  1,  1909,  a note  for  $2000  given 
July  1,  1908,  bearing  interest  at  6%  ? 

2.  I buy  a house  subject  to  a 5%  mortgage  of  $2500,  payable  half-yearly. 
What  is  the  amount  of  my  semiannual  payment? 

3.  What  semiannual  income  do  I receive  from  a $1200  bond  bearing 
interest  at  3J  % ? 

If..  A ground  rent  of  $1500  bears  interest  at  6%,  payable  quarterly  ; what  is 
the  amount  of  the  quarterly  payment? 

5.  My  property  has  against  it  a 5%  mortgage  of  $1200  and  a 6%  ground 
rent  of  $800.  What  is  the  amount  of  my  yearly  interest  payments? 

6.  Borrowed  $750  on  January  18, 1908,  at  4 per  cent.  What  amount  must 
I pay  September  5,  1908? 

7.  Bought  a house  on  August  20,  1906,  for  $8000,  paying  $2500  cash  and 
giving  a mortgage  at  5%  for  the  balance.  Paid  for  taxes,  repairs,  etc.,  $218.30. 
Collected  rents  amounting  to  $780.  Sold  the  house  September  12,  1908,  for 
$8600.  How  much  did  I gain  ? 

8.  I have  a lot  of  ground  which  I offer  for  sale  at  $1300.  How  much 
should  I charge  a man  for  it,  who  offers  to  pay  $300  down  and  settle  the  balance 
by  paying  $200  every  three  months,  in  order  to  get  6%  on  my  money? 

9.  On  March  13,  1908,  at  Philadelphia,  Pa.,  A sells  B a bill  of  goods 
amounting  to  $492.83,  on  three  months  credit.  How  much  should  B pay  A on 
October  26,  1908? 

10.  I have  an  account  with  a firm  of  bankers  and  brokers,  and  have  an 
understanding  with  them  that  they  are  to  allow  me  4 per  cent,  interest  on  the 
money  I have  on  deposit  with  them,  but,  whenever  my  account  is  overdrawn,  I 
am  to  pay  them  6 per  cent,  interest  on  the  debit  balance.  On  February  21, 1908, 
I deposit  $3216.36 ; on  April  13,  1908,  they  purchased  stock  on  my  account, 
amounting  to  $9218.75  ; on  July  7,  1908,  they  sell  the  stock  for  me  for  $9462.25. 
What  is  the  balance  to  my  credit  on  November  10,  1908? 

11.  I borrowed  $327  on  May  24,  1907,  interest  at  4f%,  and  $725.75  on  Jan- 
uary 15,  1908,  interest  at  5%.  What  was  due  on  the  two  debts  September  1, 
1908? 

12.  On  May  4,  1907,  a merchant  bought  goods  for  $780.16  on  90  days 
credit,  but  did  not  pay  for  them  until  April  7,  1908.  What  amount  did  he  pay, 
interest  being  reckoned  at  6%  from  the  expiration  of  the  90  days? 

13.  John  Smith  borrowed  $752,  June  23,  1907,  interest  at  5%,  and  $634.40 
on  December  23, 1907,  interest  at  5J%.  What  total  amount  did  he  owe  on  Octo- 
ber 1,1908? 


192 


INTEREST 


SIX  PER  CENT.  METHOD 


618.  At  six  per  cent,  the  interest  for  one  year  is  .06  of  the  principal.  For 
one  month,  or  ^ of  a year,  it  is  of  .06,  or  .005  of  the  principal.  For  one  day, 
or  of  a month,  it  is  of  .005  or  ,000|  of  the  principal. 

Example. — What  is  the  interest  of  $1348  at  6%  for  3 yr.  6 mo.  18  da.? 


3X.06  = .18  $1348 

6X.005  = .030  .213 

18  X. 000|  = .003  4044 

.213  13  48 


269  6 
$287,124 


$1348 
.033 
~ 4 044 
40  44 

$44,484  interest  for  6 mo.  18  da. 


$1348 

.06 

' 80.88 
3 

$242.64  interest  for  3 vr. 

44.48 

$287.12 


819.  Rule. — Find  what  per  cent,  of  the  principal  the  interest  will  be  for  the 
whole  time  required,  and  multiply  the  principal  by  this  decimal,  as  folloius : 

Take  as  many  times  .06  as  there  are  years,  as  many  times  .005  as  there  are 
months,  and  as  many  times  .000\  as  there  are  days  ; add  these  rates,  and  multiply  the 
principal  by  their  sum. 

Note. — Another  way  to  state  this  rule  is  : First  find  the  interest  of  one  dollar  for  the  given  time, 

and  then  take  as  many  times  this  amount  as  there  are  dollars  in  the  principal.  Or,  take  i of  the 
months  and  call  the  result  cents,  and  take  4 of  the  days  and  call  the  result  mills. 


620.  The  product  by  the  above  rule  is  the  interest  at  6%.  To  find  the 
interest  at  any  other  rate,  increase  or  diminish  this  amount  proportionally.  As, 
to  find  the  interest  at  7 %,  add  ^ ; to  find  the  interest  at  5%,  subtract  £ ; for  8%, 
add  J ; for  4%,  subtract  etc. 


ORAL  EXERCISE 

621.  1.  What  is  the  interest  of  $250  for  5 yrs.  at  6 % ? 

2.  Of  $1200  for  1 yr.  8 mo.  at  6 % ? 

3.  Of  $500  for  2 yr.  6 mo.  at  4 % ? 

4.  Of  $100  for  2 yr.  2 mo.  12  da.  at  6%  ? 

5.  Of  $800  for  1 yr.  5 mo.  15  da.  at  6 % ? 

6.  Of  $300  for  2 yr.  3 mo.  at  5 % ? 

7.  Of  $600  for  1 yr.  6 mo.  at  4 J % ? 

8.  Of  $400  for  6 mo.  10  da.  at  6%  ? 

9.  Of  $1000  for  4 mos.  at  6%  ? 

10.  Of  $200  for  1 yr.  4 mo.  at  8%  ? 


INTEREST 


193 


WRITTEN  EXERCISE 

622.  What  is  the  interest  of 

1.  $356.17  for  2 yr.  7 mo.  12  da.  at  6%  ? 

2.  $3947  for  1 yr.  5 mo.  6 da.  at  6 % ? 

3.  $758.34  for  4 yr.  8 mo.  at  6 % ? 

J.  $1263.50  for  3 yr.  4 mo.  24  da.  at  6%  ? 

5.  $478.64  for  5 yr.  9 mo.  18  da.  at  6%  ? 

6.  $592.75  for  10  mo.  15  da.  at  5%  ? 

7.  $6438.92  for  3 yr.  3 mo.  3 da.  at  7 % ? 

5“.  $1873.80  for  2 yr.  11  mo.  21  da.  at  8 % ? 

9.  $237.57  for  6 yr.  8 mo.  27  da.  at  4%  ? 

10.  $5645.20  for  1 yr.  2 mo.  2 da.  at  3%  ? 


DAY  METHOD 

623.  To  find  the  interest  of  any  principal  for  any  number  of  days  at 

6%. 


Example. — Find  the  interest  of 

June  14  days. 

July  31  “ 

Aug.  31  “ 

Sept.  7 “ 

83  days. 

June  16  to  Sept.  7 = 83  days. 

624.  Rule. — Multiply  the  principal 
pointing  off  three  places  if  the  principal  be 
and  cents. 


60  at  6%,  from  June  16  to  September  7. 

$2560 
83 
7680 
20480 
6)212480 
3541 3f 

$35.41,  interest  for  83  days. 

by  the  number  of  days,  and  divide  by  six, 
dollars,  or  five,  if  the  principal  be  dollars 


625.  In  Pennsylvania,  the  legal  rate  of  interest  being  6%  and  money 
loans  being  usually  for  terms  not  exceeding  four  months,  this  method  is  the 
basis  for  nearly  all  interest  calculations.  The  formula  is  usually  modified  as 
below  and  the  operation  shortened  by  cancelation. 

Multiply  the  principal  by  the  days,  divide  by  60  and  point  off  two  places  if 
the  principal  be  dollars  only,  four  places  if  there  be  cents  in  the  principal. 

Example. — Find  the  interest  on  $360  for  61  days  at  6%. 


6 

$^X61 

n 


= $3.66. 


This  may  be  modified  for  different  rates  by  changing  the  divisor.  For  5% 
the  divisor  is  72;  for  9%,  40;  for  4%,  90;  for4J%,  80;  and,  in  general,  the 
divisor  is  the  number  obtained  by  dividing  360  by  the  rate  per  cent. 


194 


INTEREST 


ORAL  EXERCISE 

626.  1-  What  is  the  interest  of  $600  for  20  days  at  6%? 

2.  Of  $900  for  70  days  at  6%  ? 7.  Of  $600  for  90  days  5%  ? 

3.  Of  $720  for  100  days  at  6%  ? 8.  Of  $200  for  120  days  at  3%  ? 

If..  Of  $1000  for  20  days  at  6 % ? 9.  Of  $400  for  30  days  at  7 % ? 

5.  Of  $200  for  33  days  at  4 % ? 10.  Of  $240  for  60  days  at  6%  ? 

6.  Of  $300  for  80  days  at  4|  % ? 11.  Of  $450  for  50  days  at  8%  ? 


WRITTEN  EXERCISE 

627.  Find  the  interest  of 

1.  $892  for  79  days  at  6%. 

2.  $1763  for  125  days  at  6%. 

3.  $4367  for  217  days  at  6%. 

If..  $3271.38  for  1 yr.  4 mo.  17  da.  at  6%. 

5.  $2794.72  for  2 yr.  5 mo.  23  da.  at  5%. 

6.  $745  from  March  16  to  June  19  at  6%. 

7.  $987  from  May  3 to  October  14  at  6%. 

8.  $723.93  from  August  3, 1907,  to  June  17,1908,  at  5%. 

60-DAY  METHOD 

628.  The  interest  on  any  principal  for  60  days  at  6%  is  one  per  cent,  of  the 
principal. 


Example  — Find  the  interest  of  $3564  for  95  days  at  6%. 

1 % $35.64  = 60  days  int. 

i of  60  da.  17.82  = 30  “ “ 

-jt  of  30  da.  2.97  = 5 “ “ 


$56.43  = 95  days  int. 

629.  Rule. — Find  one  per  cent,  of  the  principal  by  pointing  off  two  decimal 
places  ; this  gives  the  interest  for  60  days. 

Then  find  the  interest  for  the  required  time  by  taking  aliquot  parts  of  60-days 
interest. 


ALIQUOT  PARTS  OF  60 


630. 


5 = 

i 

1 2 

of  60 

40  = 

i 

3 

less  than  60 

63  = -W  more  than  60 

6 = 

1 

1 0 

U 

60 

45  = 

1 

4 

<< 

60 

65  = it 

60 

10  = 

1 

6 

u 

60 

48  = 

1 

T 

U 

60 

70  = i 

60 

12  = 

1 

~s 

u 

60 

50  = 

l 

6 

u 

60 

72  = i 

60 

15  = 

1 

¥ 

u 

60 

54  = 

1 

1 0 

u 

60 

75  = i 

60 

20  = 

1 

S' 

u 

60 

55  = 

1 

1 2 

a 

60 

so  = i 

60 

30  = 

1 

2 

a 

60 

57  = 

1 

2 0 

u 

60 

CO 

O 

II 

k«|M 

60 

Note. — These  numbers  may  also  be  used  as  the  basis  for  the  aliquot  parts  of  600  and  6000. 


INTEREST 


195 


ORAL  EXERCISE 

631.  1 • What  is  the  interest  of  $2374  for  60  daj'-s  at  6%  ? 

2.  Of  $1266  for  20  days  at  6%?  7.  Of  $1200  for  40  days  at  6%? 

3.  Of  $240  for  90  days  at  6%?  8.  Of  $1700  for  60  days  at  5%? 

4.  Of  $720  for  65  days  at  6%?  9.  Of  $600  for  70  days  at  7 %? 

5.  Of  $3000  for  50  days  at  3%?  10.  Of  $300  for  80  days  at  8 %? 

6.  Of  $1500  for  30  days  at  6%?  11.  Of  $1700  for  93  days  at  4J%? 


WRITTEN  EXERCISE 

632.  Find  the  interest  of 

1.  $437.56  for  57  days  at  6fi. 

2.  $5278  for  66  days  at  6%. 

3.  $726.32  for  8 mo.  20  da.  at  6%. 

4.  $3258  for  1 yr.  2 mo.  18  da.  at  4%. 

5.  $493.72  for  2 yr.  7 mo.  16  da.  at  6%. 

6.  $1267.50  for  183  days  at  6%. 

7.  $2784.10  for  9 mo.  8 da.  at  7%. 

8.  $521.56  for  1 yr.  5 mo.  23  da.  at  8%. 
$972.83  for  231  days  at  5%. 

10.  $6238  for  7 mo.  29  da.  at  6%. 


“ $6000  RULE  ” 

633.  The  interest  of  $6000  at  6%  is  $1  a day. 

Example. — What  is  the  interest  of  $8000  at  6 % for  210  days  ? 

Interest  of  $6000  for  210  days  = $210 

“ “ $2000  (i  of  $6000)  = 70 

“ “ $8000  for  210  days  = $280 

ORAL  EXERCISE 

634.  1.  What  is  the  interest  of  $3000  for  280  days  at  6%? 

2.  Of  $500  for  72  days  at  6%? 

3.  Of  $2000  for  315  days  at  6%? 

4.  Of  $9000  for  120  days  at  7 %? 

5.  Of  $7500  for  84  days  at  4%? 


WRITTEN  EXERCISE 


635.  Find  the  interest  of 

1.  $3000  for  2 yr.  3 mo.  18  da.  at  4%. 

2.  $2400  for  1 yr.  9 mo.  16  da,  at  6%. 

3.  $4500  for  3 yr.  4 mo.  22  da.  at  6%. 

4 ■ $1200  for  1 yr.  1 mo.  3 da.  at  2%. 

5.  $12500  for  2 yr.  7 mo.  8 da.  at  5%. 


6.  $1000  for  172  days  at  6%. 

7.  $1500  for  312  days  at  6%. 

8.  $2000  for  1 yr.  27  da.  at  6% 

9.  $4000  for  278  days  at  7 %. 
10.  $9600  for  78  days  at  6%. 


196 


INTEREST 


Common  Year — 360  Days 

636.  The  interest  on  $1  for  360  days  at  1%,  or  on  |360  for  1 day  at  1%  is 
one  cent.  In  proportion  as  the  rate  is  increased,  the  time  or  principal  to  yield 
one  cent  of  interest  is  decreased.  Thus  $1  at  3%,  4%,  or  6%,  will  yield  one  cent 
of  interest  in  respectively  120,  90,  60  days,  or  on  respectively  $120,  $90,  or  $60 
for  one  day.  This  applies  to  any  rate  divisible  evenly  into  360. 

Note. — In  the  illustrative  table  it  will  be  seen  that  pointing  off  two  places  in  the  principal 
gives  the  interest  for  as  many  days  as  the  rate  is  contained  times  in  360  ; or,  conversely,  pointing  off 
two  places  in  the  days  gives  the  interest  on  a like  number  of  dollars.  Also,  that  pointing  off  ODe 
place  gives  the  interest  for  ten  times  as  many  days  or  on  ten  times  as  many  dollars  ; while  three  places 
pointed  off  gives  the  interest  for  one-tenth  as  many  days  or  on  one-tenth  as  many  dollars. 


Illustrative  Table 


$1  at  2 % for  180  days,  or  $180  at  2 

% for  1 day  $.01 

$1  at  3 % 

“ 120 

“ or  $120  at  3 

% “ “ “ 

.01 

$1  at  6 % 

“ 60 

“ or  $60  at  6 

% “ “ “ 

.01 

$1  at  4 % 

“ 90 

“ or  $90  at  4 

% “ “ “ 

.01 

$1  at  4i% 

“ 80 

“ or  $80  at  4J%  “ “ “ 

.01 

$1  at  5 % 

“ 72 

“ or  $72  at  5 

% “ “ “ 

.01 

$1  at  6 % 

“ 6 

“ or  $6  at  6 

% “ “ “ 

.001 

$1  at  6 % 

“ 600 

“ or  $600  at  6 

% “ “ “ 

.10 

Example  1. — Find  interest  on  $720  at  4%  for  135  days. 

(1)  '(2) 

$7.20  int.  for  90  days  $1.35  int.  on  $90 

_3.60  “ “ 45  “ (|  of  90)  8 8 

$10.80  ““  135  “ $10.80  “ “$720  (90X8) 


Example  2. — Find  interest  on  $960  at  4|%  for  64  days. 

(1)  **  (2) 

$9.60  int.  for  80  days  .64  int.  on  $80 

1.92  “ “ 16  “ Q- less)  .12 12 

$7.68  “ “ 64  “ $7.68  $960  (80X12) 


Example  3. — Find  the  interest  on  $780  for  19  days  at  6%. 


$.780 

3 

2.340 

130 


(!) 

int.  6 days 
3 

“ 18  “ (6X3) 


(2) 

$.019  int.  on  $6 

130  130 

$2,470  $780  (6X130) 


$2.4700  “ 19 


u 


INTEREST 


197 


WRITTEN  EXERCISE 

638.  1 . Find  the  interest  on  $475.84  for  96  days  at  5%. 

2.  What  is  the  interest  on  $286.80  for  126  days  at  6%  ? 

3.  Find  the  interest  on  $876.92  for  69  days  at  8%. 

I/..  What  is  the  interest  on  $345.75  for  136  days  at  4|%  ? 

5.  Find  the  interest  on  $595  for  288  days  at  4%. 

6.  What  is  the  interest  on  $1350  for  298  days  at  4%  ? 

7.  Find  the  interest  on  $330  for  129  days  at  6%. 

8.  What  is  the  interest  on  $1320  for  217  days  at  5%  ? 

REVIEW  PROBLEMS  IN  SIMPLE  INTEREST 

In  the  problems  given  below  let  the  student  use  the  special  rule  that  seems  best  suited  to  the 
problem  to  be  worked.  In  general,  we  suggest  that  for  terms  extending  beyond  a year  the  time  be 
expressed  in  years,  months  and  days,  and  that  the  interest  be  found  for  the  years  and  aliquot  parts  of 
a year.  For  terms  of  less  than  a year,  let  the  time  be  expressed  in  days,  and  the  day  method,  or  its 
modification,  the  60-day  method  be  used. 

639.  1 ■ Find  the  interest  on  $4500  for  1 yr.  1 mo.  1 da.  at  6%. 

2.  What  is  the  amount  of  $2250  for  3mo.  3 da.  at  6%  ? 

3.  Find  the  interest  on  $405.50  for  66  days  at  7%. 

If.  What  is  the  interest  on  $1606  from  March  2,  1908,  to  May  17,  1909, 
at  4%? 

5.  John  Hamaker  took  up  his  note  for  $350  to-day,  January  27,  1909, 
which  was  dated  April  1,  1908,  and  bore  interest  at  5%  ; what  amount  did 
he  pay  ? 

6.  What  is  the  interest  on  $778.75  at  6%  from  September  1, 1908,  to  Janu- 
ary 1,  1909? 

7.  Find  the  amount  of  $500  for  5 mo.  5 da.  at  5%. 

8.  What  is  the  interest  on  $1755  for  1 yr.  9 mo.  9 da.  at  6%  ? 

9.  What  is  the  interest  on  $299.30  for  192  days  at  7 % ? 

10.  What  is  the  interest  on  $1666  for  6 mo.  6 da.  at  6 % ? 

11.  What  is  the  interest  on  $777  for  77  days  at  7 % ? 

12.  Find  the  amount  of  $2200  from  January  1,  1908,  to  September  1,  1908, 
at  4J%. 

13.  What  is  the  interest  on  $920  for  8 mo.  8 da.  at  8%  ? 

Ilf.  What  is  the  amount  of  $444  for  44  days  at  4%  ? 

15.  What  is  the  interest  on  $333  for  3 yr.  3 mo.  3 da.  at  34%  ? 

16.  Find  the  interest  on  $508.50  for  11  mo.  15  da.  at  9%  ? 

17.  Find  the  interest  on  $1532.75  from  February  1,  1908,  to  October  1,  1908, 
at  10%. 

18.  What  is  the  amount  of  $1410.18  for  10  mo.  26  da.  at  12%  ? 

19.  Sept.  4,  1907,  I bought  a quantity  of  grain  for  $840.70;  Nov.  7,  I sold 
part  of  it  for  $580.90;  Jan.  27,  1908,  the  remainder  for  $325.40.  Money  being 
worth  6%,  how  much  did  I gain  by  the  transaction? 

20.  May  9,  1907,  a dealer  borrowed  sufficient  money  at  5 % to  pay  for  280 
bushels  corn,  at  65c.  a bushel  and  for  300  bushels  oats  at  40c.  a bushel.  Corn 
advanced  8c.  a bushel,  and  oats  4c.  a bushel.  He  sold  at  these  advanced  prices 
on  July  27.  After  returning  the  money  borrowed  and  the  accrued  interest,  what 
was  his  gain  ? 


198 


INTEREST 


ACCURATE  INTEREST 

640.  Accurate  Interest  is  found  by  taking  the  exact  number  of  days,  and 
reckoning  them  as  365tbs  of  a year. 

Example. — What  is  the  accurate  interest  of  $721  at  5%  for  213  days? 


.0/ 


7-2  / 7 2/ X.SS  X 2 /3 

.0  S 3~£S 

3 • O 3 7*3  2 / 3 

2 / 3 


72  / 

2 / .3 


/ o f / s 

3 7 0 S 
7 2/Q 

3 7 SJf  7 C,  7 f.  6 s (/ 2 /.  037 
73  0 K 

37 r 

3 3S 


/ S / S / 

73)  / S3  S.73/2  /.037 
/ V 6,  K 
7S 

JIA 


2 7J? 
2/0 


/ 3 7 S 
/ 0 7 s 


S+/0 


2 7 0 0 


Note. — In  ordinary  interest  a year  consists  of  360  days  ; in  accnrate  interest  it  consists  of  365 
days.  Since  360  days  is  or  7aj  less  than  365  days,  it  follows  that  the  interest  for  a year  of  360  days 
is  A-  more  than  the  interest  for  a year  of  365  days  ; hence  accurate  interest  may  be  reckoned  by  first 
finding  the  ordinary  interest  and  then  deducting  ^ of  this  amount. 


641.  Rule. — Find  the  interest  for  one  year  by  multiply iny  the  principal  by  the 
rate  expressed  decimally. 

Multiply  the  interest  for  one  year  by  the  number  of  days,  and  divide  the  product 
by  365. 


WRITTEN  EXERCISE 

642.  Find  the  accurate  interest  of 

1.  $623  for  26  days  at  6%. 

2.  $937.25  for  123  days  at  5%. 

3.  $685  for  73  days  at  4%. 

J.  $427.60  for  92  days  at  6%. 

5.  $1272.50  for  34  days  at  3%. 

6.  $561  from  Jan.  13,  1908,  to  June  3,  190S,  at  7%. 

7.  $743.25  from  April  17,  1908,  to  Aug.  13,  190S,  at  5%. 

8.  $1864.73  from  Feb.  1,  1908,  to  Dec.  21,  1908,  at  6%. 

9.  $324  from  April  20,  1908,  to  Oct.  5,  1908,  at  2J%. 

10.  $911.75  from  June  27,  1908,  to  Nov.  5,  1908,  at  44%. 


INTEREST 


199 


INTEREST  PROBLEMS 

643.  To  find  the  rate,  when  the  principal,  interest  and  time  are 
given. 

Example. — At  what  rate  will  $480  produce  $4.50  interest  in  75  days? 

$480  principal. 

$4.80  int.  for  60  days  at  6%. 

1-20  “ “ 15  “ “ “ (4  of  60  days). 

$6.00  “ “ 75  “ “ “ 

"fLOO  “ “ 75  “ “1%  (iof6%). 

$4.50-?-$l=4J  Result  44%. 

644.  Rule. — Divide  the  given  interest  hy  the  interest  of  the  given  principal  at 
1 °/P  for  the  given  time. 

MENTAL  PROBLEMS 

645.  1.  At  what  rate  per  cent,  will  $100  produce  $6  interest  in  1 year? 

Solution. — As  1 per  cent.,  which  is  of  the  principal,  will  produce  $1  in  one  year,  to  produce 

$6  will  require  as  many  times  1 per  cent,  as  $1  is  contained  times  in  $6,  or  6 per  cent. 

2.  At  what  rate  per  cent,  will  $200  produce  $12  in  1 year? 

3.  At  what  rate  per  cent,  will  $300  produce  $20  in  2 years  ? 

If..  Find  what  rate  per  cent,  will  produce  $7  interest  on  $200  in  6 months. 

5.  I received  $3  interest  by  investing  $180  for  60  days ; what  rate  per  cent, 
was  paid  ? 

6.  Having  $200  on  interest  for  3 years,  I received  principal  and  interest, 
$236  ; what  rate  per  cent,  did  I get  for  my  money  ? 

7.  A loan  of  $300,  which  had  been  out  for  4 years,  was  returned  with  $60 
interest ; what  was  the  rate  per  cent.  ? 

8.  At  what  rate  per  cent,  will  $160  in  5 years  give  $64  interest  ? 

9.  Find  at  what  rate  per  cent.  $100  will  in  4 years  amount  to  $128. 

10.  At  what  rate  per  cent,  will  $500  amount  to  $510  in  90  da)rs? 

WRITTEN  EXERCISE 

646.  At  what  rate  will 

1.  $725  produce  $8.82  interest  in  73  days? 

2.  $2430  produce  $25.92  interest  in  6 mo.  12  da.? 

3.  $718.75  produce  $39.53  interest  in  1 yr.  2 mo.  20  da.? 

If.  $522.50  produce  $96.41  interest  in  2 yr.  7 mo.  19  da.? 

5.  $358.60  produce  $9.40  interest  in  5 mo.  27  da.  ? 

6.  $2332  produce  $282.43  interest  in  3 yr.  10  da.? 

7.  $2765.10  produce  $230.23  interest  in  1 yr.  6 mo.  5 da.  ? 

8.  $563.25  produce  $23.31  interest  in  213  days  ? 

9.  $917  amount  to  $989.81  in  2 yr.  5 mo.  17  da.? 

10.  $3223.96  amount  to  $3250.60  in  10  mo.  25  da.? 


200 


INTEREST 


647.  To  find  the  time,  when  the  principal,  interest  and  rate  are  given. 

Example. — In  what  time  will  $375  amount  to  $412.38,  at  5%  per  annum? 


$412.38  amt. 
375.00  prin. 

37.38  int. 


1 

9936 

18.75  )37.38 
18  75 

0000 

or, 


$375 
.05 

$18.75  int.  for  1 yr. 

18.75  ) 37.38000  ( 1.9936  yr. 

12 


18  630 
16  875 

1 7550 
1 6875 

6750 

5625 

11250 


18  75 

18  630 
16  875 

1 7550 
1 6875 

~ 6750 
5625 


11.9232  mo. 
30 

27.6960 


11250 

Result  1 yr.  11  mo.  28  da. 


648.  Rule. — Divide  the  given  interest  by  the  interest  of  the  given  principal  for 
one  year  at  the  given  rate.  The  result  will  be  years ; reduce  the  fraction,  if  any,  to 
months  and  days. 

MENTAL  EXERCISE 

649.  1.  In  what  time  will  $100  at  6 percent,  earn  $18  interest  ? 

Solution. — In  1 year  the  interest  will  equal  T or  of  $100,  which  is  $6,  and  to  earn  $18  will 
require  as  many  years  as  $6  is  contained  times  in  $18,  or  3 years. 


2.  In  what  time  will  $200  at  5 per  cent,  earn  $50  interest? 

3.  In  what  time  will  $200  at  7 per  cent,  earn  $28  interest? 

If..  In  what  time  will  $200  at  6 per  cent,  earn  $2  interest  ? 

5.  In  what  time  will  $300  at  10  per  cent,  earn  $90  interest? 

6.  In  what  time  will  $80  at  5 per  cent,  earn  $10  interest? 

7.  In  what  time  will  $300  at  6 per  cent,  amount  to  $309  ? 

8.  In  what  time  will  $50  at  6 per  cent,  earn  $1J  interest? 


WRITTEN  EXERCISE 

650.  1.  In  what  time  will  $5000  produce  $112.70  at  6%  ? 

2.  $2230  produce  $64.27  at  6%  ? 

3.  $3917.25  produce  $910  at  5%  ? 
f.  $325  produce  $72.18  at  4%  ? 

5.  $1897.10  produce  $321.37  at  3%  ? 

6.  $376.75  produce  $1S4.12  at  8%  ? 

7.  $632.80  amount  to  $711.13  at  6%  ? 

8.  $421.89  amount  to  $500  at  54%  ? 

9.  $1124  produce  $321.17  at  2 J%  ? 

10.  $789.73  amount  to  $821.50  at  7 % ? 

11.  In  what  time  will  $4263.13  produce  $289.10  interest  at  54%  per  annum? 

12.  In  what  time  will  $750  produce  $150  interest  at  54  %? 


INTEREST 


201 


651.  To  find  the  principal,  when  the  interest,  time  and  rate  are 
given. 

Example. — What  principal  will  produce  $42.99  interest  in  2 yr.  3 mo.  15 
da.  at  5^>? 

$1  .1145f)42.9900( 

.05  6 6 


$.05  int.  for  1 yr. 

.10  int.  for  2 yr. 

.0125  int.  for  3 mo.  of  1 yr.) 
.0020f-  int.  for  15  da.  of  3 mo.) 
$.1145f  int.  for  2 yr.  3 mo.  15  da. 


Result  $375.19 


.6875  )257.9400(  375.186 
206  25 
51  690 
48125 
3 5650 
3 4375 
12750 
6875 
58750 
55000 
37500 


652.  Rule. — Divide  the  given  interest  by  the  interest  of  $ 1 at  the  given  rate  for 
the  given  time. 

653.  To  find  the  principal,  when  the  amount,  time  and  rate  are 
given. 

Example. — What  principal  will  amount  to  $594.04  in  2 yr.  3 mo.  18  da.  at 

6%  ? 

$1  1.138)594.040(522 

.06 

$ 06  int.  for  1 yr. 

.12  int.  for  2 yr. 

.015  int.  for  3 mo. 

.003  int.  for  18  da.  $1,138  amt.  of  $L00. 

$.138  int.  for  2 yr.  3 mo.  18  da. 

654.  Rule. — Divide  the  given  amount  by  the  amount  of  $1  at  the  given  rate  for 
the  given  time. 


569  0 
25  04 
22  76 
2 280 
2 276 

4 Result  $522. 


MENTAL  EXERCISE 

655.  1.  What  principal  will  in  two  years,  at  6 per  cent.,  amount  to  $112  ? 

Solution. — At  6 per  cent,  for  2 years  TW  of  the  principal  equals  the  interest.  plus  TVo  or 
ijg  equals  the  amount  or  $112,  and  x^o  equals  xj2  of  the  amount  or  $1;  hence  the  principal  is  $100. 

2.  What  principal  will  in  7 years,  at  5 per  cent.,  amount  to  $270? 

3.  What  principal  will  in  4 years,  at  10  per  cent.,  amount  to  $140? 


202 


INTEREST 


1/..  AVhat  principal  will  in  3J  years,  at  6 per  cent.,  amount  to  $242  ? 

5.  What  principal  will  in  2 years,  at  4 per  cent.,  produce  $120  interest? 

6.  What  principal  will  in  1 year  6 months,  at  7 per  cent.,  amount  to  $221  ? 

7.  What  principal  will  in  6 months,  at  6 per  cent.,  amount  to  $412? 

8.  What  principal  will  in  60  days,  at  6 percent.,  produce  $6  interest? 

9.  What  principal  will  in  3 months,  at  12  per  cent.,  amount  to  $206  ? 

10.  What  sum  of  money  was  put  at  interest  3 years  ago  at  5 per  cent.,  if  it 
now  amounts  to  $575  ? 

11.  What  principal,  put  at  interest  at  8 per  cent,  for  2 years  and  6 months, 
will  produce  $60  interest  ? 

12.  What  principal,  put  at  interest  for  120  days  at  6 per  cent.,  will  produce 
$16  interest  ? 

13.  What  principal,  put  at  interest  for  3 years  at  5 per  cent.,  will  amount  to 
$805? 

14-.  What  principal,  put  at  interest  for  30  days  at  6 per  cent.,  will  produce  $6 
interest  ? 

15.  I received  to-day  $1080,  which  is  the  amount  of  a sum  of  money 
piaced  on  interest  for  2 years  at  4 per  cent.;  what  was  the  sum  ? 

WRITTEN  EXERCISE 

656.  What  principal  will  produce 

1.  $37.27  interest  in  179  days  at  6%  ? 

2.  $184.29  interest  in  1 yr.  3 mo.  3 da.  at  4%  ? 

3.  $274.50  interest  in  8 mo.  21  da.  at  6%  ? 

4 $381.75  interest  in  175  days  at5%? 

5.  $92.17  interest  in  2 yr.  7 mo.  11  da.  at  6%  ? 

6.  $188.77  interest  in  5 mo.  12  da.  at  7 % ? 

7.  $941.15  interest  in  11  mo.  18  da,  at  34%  ? 

8.  $62.75  interest  in  3 yr.  1 mo.  27  da.  at  6%  ? 

9.  $713.41  interest  in  4 mo.  18  da.  at  2%  ? 

10.  $825  interest  in  1 yr.  10  mo.  2 da.  at  54%  ? 

What  principal  will  amount  to 

11.  $4692.10  in  7 mo.  8 da.  at  6%  ? 

12.  $378.12  in  2 yr.  1 mo.  5 da.  at  4%  ? 

13.  $927.38  in  1 yr.  7 mo.  4 da.  at  7 % ? 

11/..  $1250  in  5 mo.  23  da,  at  6 % ? 

15.  $3565.15  in  11  mo.  14  da.  at  8%  ? 

16.  $2732.44  in  3 yr.  8 mo.  at  5%  ? 

17.  $5723.92  in  4 yr.  2 mo.  7 da.  at  4 4%  ? 

18.  $125.78  in  10  mo.  28  da.  at  3%  ? ~ 

19.  $3716.17  in  1 yr.  3 mo.  13  da.  at  2%  ? 

20.  $2244.39  in  2 yr.  4 mo.  24  da.  at  6%  ? 


COMPOUND  INTEREST 


657.  Compound  Interest  is  interest  on  the  principal,  and  on  the  unpaid 
interest  after  it  becomes  due. 


Note. — Compound  interest  will  not  be  allowed  legally  unless  there  has  been  an  account  settled 
between  the  parties,  or  a judgment  has  been  obtained,  thus  producing  a new  principal,  aggregating 
principal  and  interest  which  has  fallen  due,  or  where  there  is  a special  agreement  in  valid  form  to  pay 
compound  interest. 

It  will  also  be  allowed  upon  coupons  which  are  overdue,  or  upon  interest  wrongfully  withheld 
by  a trustee  from  a beneficiary. 


Example. — Find  the  compound  interest  of  $650  for  3 yr.  7 mo.  12  days 
at  6%. 

Explanation. — Since  the  interest  is  to  be 
compounded  annually,  the  amount  due  at  the  end 
of  the  first  year,  which  is  $689,  will  be  the  basis  of 
interest  for  the  second  year  ; and  the  amount  due 
at  end  of  second  year,  $730.34,  will  be  the  basis  of 
interest  for  the  third  year  ; the  amount  due  at 
the  end  of  3d  year,  $774.16,  will  be  the  basis  of 
the  interest  for  the  remaining  7 mo.  12  days  of 
the  time  ; and  since  the  compound  amount  thus 
found,  $802.80,  is  made  up  of  the  compound  inter- 
est and  the  principal,  if  from  this  amount  the 
principal  be  subtracted,  the  remainder,  $152.80, 
will  be  the  compound  interest. 


$650 

prin. 

39. 

int.  for  1st  yr. 

689. 

amt.  at  end  of  1st 

yr. 

4134 

int.  for  2nd  yr. 

730.34 

amt.  at  end  of  2d 

yr- 

43.82 

int.  for  3d  yr. 

774.16 

amt.  at  end  of  3d 

yr. 

28.64 

int.  for  7 mo.  12  days. 

802.80 

amt.  for  full  time. 

650.00 

prin. 

$152.80 

compound  int. 

658.  Rule. — Find,  the  amount  of  the  principal  for  the  first  period  of  time  for 
which  interest  is  to  be  reckoned,  and  make  this  the  principal  for  the  second.  Find  the 
amount  of  this  principal  for  the  second  period,  and  thus  continue  to  the  end  of  the 
given  time.  The  last  amount  will  be  the  required  amount.  To  find  the  compound 
interest,  subtract  the  given  principal  from  the  last  amount. 

Note. — If  the  time  contains  fractional  parts  of  a period,  as  months  and  days,  find  the  amount 
due  for  the  full  periods,  and  to  this  add  its  interest  for  the  months  and  days. 

Remarks. — 1.  For  interest  to  be  compounded  semiannually,  take  one-half  the  rate  for  twice  the 

time. 

2.  For  interest  to  be  compounded  quarterly,  take  one-fourth  the  rate  for  four  times  the  time. 

3.  For  interest  to  be  compounded  bi-monthly,  take  one-sixth  the  rate  for  six  times  the  time. 

4.  For  periods  beyond  the  scope  of  the  table,  multiply  together  the  amounts  shown  for  periods, 
the  sum  of  which  will  equal  the  time  required.  Thus,  the  amount  of  $1  at  compound  interest  for  65 
years  at  4 %,  is  equal  to  the  product  of  $1  at  4%  for  30  years,  and  the  amount  of  $1  at  4%  for  35  years, 
that  is,  3.24339751  X 3.94608899,  or  12.79873520. 


WRITTEN  RROBLEMS 

659.  1.  What  is  the  compound  interest  of  $340  for  3 years  at  5 % ? 

£.  What  is  the  compound  interest  of  $750  for  2 years  6 mouths  at  6%,  pay- 
able semiannually  ? 

3.  I lent  a contractor  $5400  for  7 years  at  10%  per  annum,  interest 
payable  quarterly,  and  took  a bond  and  mortgage  to  secure  the  debt  and  its 
interest.  Not  having  been  paid  until  the  end  of  the  7 years,  how  much  was 
required  in  full  settlement  ? 


203 


204 


INTEREST 


If.  Aug.  1,  190S,  I paid  in  full  a note  for  $1275,  dated  April  18,  1903, 
bearing  10%  interest.  If  the  interest  was  compounded  annually,  what  was  the 
amount  due  at  settlement  ? 

5.  Money  being  worth  6%,  compounded  semiannually,  which  would  be  the 
better,  and  how  much,  for  a capitalist  to  lend  $20000  for  5 years  and  6 months, 
or  to  invest  it  in  land  that,  at  the  end  of  the  time  named,  will  sell  for  $50000 
above  all  expenses  for  taxes? 

6.  At  what  rate  per  annum  will  $5000  amount  to  $20580.678,  if  compounded 
annually  for  29  years  ? 

7.  At  what  rate  per  annum  will  $3500  amount  to  $8007.74688,  if  com- 
pounded semiannually  for  14  years  ? 

8.  A invested  $3000  at  7 % when  he  was  25  years  of  age.  How  old  was  he 
when  the  investment,  with  its  interest  compounded  semiannually,  amounted  to 
$16754.78? 

9.  If  compounded  annually  at  5%,  what  principal  will  amount  to  $2626.674 
in  34  years  ? 

10.  If  I deposit  $275  in  a savings  bank,  and  the  interest  thereon  is  com- 
pounded semiannually  at  6%  per  annum,  how  much  should  I receive  at  the  end 
of  20  years,  if  nothing  has  been  previously  withdrawn  ? 

11.  Mr.  Smith  borrowed  $2400  on  Apr.  1,  1900,  at  8%  compound  interest, 
payable  quarterly.  What  sum  was  due  Feb.  1,  1908,  if  the  first  five  payments 
were  made  when  due,  and  no  subsequent  payments  were  made? 

12.  Afather  invested  $3000  at  6%  per  annum,  interest  payable  semiannually, 
and  on  the  same  terms  promptly  invested  the  interest  as  collected.  How  much 
should  his  son  receive,  when  he  attains  his  majority,  if  he  was  6 years  old  when 
the  investment  was  made? 

13.  Find  the  amount  of  $575.80  in  6 years  at  5%  compound  interest. 

Ilf.  What  is  the  compound  interest  of  $672  for  3 years  at  4%  payable  semi- 
annually ? 

15.  What  principal  will  in  4 years  produce  $180.40  compound  interest 
at  6%  ? 


COMPOUND  INTEREST  TABLE 


205 


680.  The  labor  of  computing  compound  interest  may  be  greatly  shortened 
by  the  use  of  the  following 

Compound  Interest  Table 

Showing  the  amount  of  $1  at  compound  interest  at  various  rates  per  cent, 
for  any  number  of  years,  from  1 year  to  50  years,  inclusive: 


Yrs. 

i per  ct. 

1 1-2  per  ct. 

2 per  ct. 

2 i-2  per  ct.  3 per  ct. 

3 1-2  per  ct. 

4 per  ct. 

1 

1.0100  000 

1.0150  000 

1.0200  0000 

1.0250  0000  1.0300  0000 

1.0350  0000 

1.0400  0000 

2 

1.0201  000 

1.0302  250 

1.0404  0000 

1.0506  2500  1.0609  0000 

1.0712  2500 

1.0816  0000 

3 

1.0303  010 

1.0456  784 

1.0612  0800 

1.0768  9062  1.0927  2700 

1.1087  1787 

1.1248  6400 

4 

1.0406  040 

1.0613  636 

1.0824  3216 

1.1038  1289  1.1255  0881 

1.1475  2300 

1.1698  5856 

5 

1.0510  101 

1.0772  840 

1.1040  8080 

1.1314  0821  1.1592  7407 

1.1876  8631 

1.2166  5290 

6 

1.0615  202 

1.0934  433 

1.1261  6242 

1.1596  9342  1.1940  5230 

1.2292  5533 

1.2653  1902 

7 

1.0721  354 

1.1098  450 

1.1486  8567 

1.1886  8575  1.2298  7387 

1.2722  7926 

1.3159  3178 

8 

1.0828  567 

1.1264  926 

1.1716  5938 

1.2184  0290  1.2667  7008 

1.3168  0904 

1.3685  6905 

9 

1.0936  853 

1.1433  900 

1.1950  9257 

1.2488  6297  1.3047  7318 

1.3628  9735 

1.4233  1181 

10 

1.1046  221 

1.1605  408 

1.2189  9442 

1.2800  8454  1.3439  1638 

1.4105  9876 

1.4802  4428 

11 

1.1156  683 

1.1779  489 

1.2433  7431 

1.3120  8666  1.3842  3387 

1.4599  6972 

1.5394  5406 

12 

1.1268  250 

1.1956  182 

1.2682  4179 

1.3448  8882  1.4257  6089 

1.5110  6866 

1.6010  3222 

13 

1.1380  933 

1.2135  524 

1.2936  0663 

1.3785  1104  1.4685  3371 

1.5639  5606 

1.6650  7351 

14 

1.1494  742 

1.2317  557 

1.3194  7876 

1.4129  7382  1.5125  8972 

1.6186  9452 

1.7316  7645 

15 

1.1609  690 

1.2502  321 

1.3458  6834 

1.4482  9817  1.5579  6742 

1.6753  4883 

1.8009  4351 

16 

1.1725  786 

1.2689  855 

1.3727  8570 

1.4845  0562  1.6047  0644 

1.7339  8601 

1.8729  8125 

17 

1.1843  044 

1.2880  203 

1.4002  4142 

1.5216  1826  1.6528  4763 

1.7946  7555 

1.9479  0050 

18 

1.1961  475 

1.3073  406 

1.4282  4625 

1.5596  5872  1.7024  3306 

1.8574  8920 

2.0258  1652 

19 

1.2081  090 

1.3269  507 

1.4568  1117 

1.5986  5019  1.7535  0605 

1.9225  0132 

2.1068  4918 

20 

1.2201  900 

1.3468  550 

1.4859  4740 

1.6386  1644  1.8061  1123 

1.9897  8886 

2.1911  2314 

21 

1.2323  919 

1.3670  578 

1.5156  6634 

1.6795  8185  1.8602  9457 

2.0594  3147 

2.2787  6807 

22 

1.2447  159 

1.3875  637 

1.5459  7967 

1.7215  7140  1.9161  0341 

2.1315  1158 

2.3699  1879 

23 

1.2571  630 

1.4083  772 

1.5768  9926 

1.7646  1068  1.9735  8651 

2.2061  1448 

2.4647  1555 

24 

1.2697  346 

1.4295  028 

1.6084  3725 

1.8087  2595  2.0327  9411 

2.2833  2849 

2.5633  0417 

25 

1.2824  320 

1 . 4509  454 

1.6406  0599 

1.8539  4410  2.0937  7793 

2.3632  4498 

2.6658  3633 

26 

1.2952  563 

1.4727  095 

1.6734  1811 

1.9002  9270  2.1565  9127 

2.4459  5856 

2.7724  6979 

27 

1.3082  089 

1.4948  002 

1.7068  8648 

1.9478  0002  2.2212  8901 

2.5315  6711 

2.8833  6858 

28 

1.3212  910 

1.5172  222 

1.7410  2421 

1.9964  9502  2.2879  2768 

2.6201  7196 

2.9987  0332 

29 

1.3345  039 

1.5399  805 

1.7758  4469 

2.0464  0739  2.3565  0551 

2.7118  7798 

3.1186  5145 

30 

1.3478  490 

1.5630  802 

1.8113  6158 

2.0975  6758  2.4272  6247 

2.8067  9370 

3.2433  9751 

31 

1.3613  274 

1.5865  264 

1.8475  8882 

2.1500  0677  2.5000  8035 

2.9050  3148 

3.3731  3341 

32 

1.3749  407 

1.6103  243 

1.8845  4059 

2.2037  5694  2.5750  8276 

3.0067  0759 

3.5080  5875 

33 

1.3886  901 

1.6344  792 

1.9222  3140 

2.2588  5086  2.6523  3524 

3.1119  4235 

3.6483  8110 

34 

1.4025  770 

1.6589  964 

1.9606  7603 

2.3153  2213  2.7319  0530 

3.2208  6033 

3.7943  1634 

35 

1.4166  028 

1.6838  813 

1.9998  8955 

2.3732  0519  2.8138  6245 

3.3335  9045 

3.9460  8899 

36 

1.4307  688 

1.7091  395 

2.0398  8734 

2.4325  3532  2.8982  7833 

3.4502  6611 

4.1039  3255 

37 

1.4450  765 

1.7347  766 

2.0806  8509 

2.4933  4870  2.9852  2668 

3.5710  2543 

4.2680  8986 

38 

1.4595  272 

1.7607  983 

2.1222  9879 

2.5556  8242  3.0747  8348 

3.6960  1132 

4.4388  1345 

39 

1.4741  225 

1.7872  103 

2.1647  4477 

2.6195  7448  3.1670  2698 

3.8253  7171 

4.6163  6599 

40 

1.4888  637 

1.8140  184 

2.2080  3966 

2.6850  6384  3.2620  3779 

3.9592  5972 

4.8010  2063 

41 

1.5037  524 

1.8412  287 

2.2522  0046 

2.7521  9043  3.3598  9893 

4.0978  3381 

4.9930  6145 

42 

1.5187  899 

1.8688  471 

2.2972  4447 

2.8209  9520  3.4606  9589 

4.2412  5799 

5.1927  8391 

43 

1.5339  778 

1.8968  798 

2.3431  8936 

2.8915  2008  3.5645  1677 

4.3897  0202 

5.4004  9527 

44 

1.5493  176 

1.9253  330 

2.3900  5314 

2.9638  0808  3.6714  5227 

4.5433  4160 

5.6165  1508 

! 45 

1.5648  107 

1.9542  130 

2.4378  5421 

3.0379  0328  3.7815  9584 

4.7023  5855 

5.8411  7568 

46 

1.5804  589 

1.9835  262 

2.4866  1129 

3.1138  5086  3.8950  4372 

4.8669  4110 

6.0748  2271 

47 

1.5962  634 

2.0132  791 

2.5363  4351 

3.1916  9713  4.0118  9503 

5.0372  8404 

6.3178  1562 

, 48 

1.6122  261 

2.0434  783 

2.5870  7039 

3.2714  8956  4.1322  5188 

5.2135  8898 

6.5705  2824 

1 49 

1.6283  483 

2.0741  305 

2.6388  1179 

3.3532  7680  4.2562  1944 

5.3960  6459 

6.8333  4937 

50 

1 

1.6446  318 

2.1052  424 

2.6915  8803 

3.4371  0872  4.3839  0602 

5.5849  2686 

7.1066  8335 

206 


COMPOUND  INTEREST  TABLE 


Compound  Interest  Table 

Showing  the  amount  of  $1  at  compound  interest,  at  various  rates  per  cent, 
for  any  number  of  years,  from  1 year  to  50  }rears,  inclusive. 


Yrs. 

4 1-2  per  ct. 

5 per  ct. 

6 per  ct. 

7 per  ct. 

8 per  ct. 

9 per  ct.  10  per  ct. 

1 

1.0450  0000 

1.0500  000 

1.0600  000 

1.0700  000 

1.0800  000 

1.0900  000  1.1000  000 

2 

1.0920  2500 

1.1025  000 

1.1236  000 

1.1449  000 

1.1664  000 

1.1881  000  1.2100  000 

3 

1.1411  6612 

1.1576  250 

1.1910  160 

1.2250  430 

1.2597  120 

1.2950  290  1.3310  000 

4 

1.1925  1860 

1.2155  063 

1.2624  770 

1.3107  960 

1.3604  890 

1.4115816  1.4641000 

5 

1.2461  8194 

1.2762  816 

1.3382  256 

1.4025  517 

1.4693  281 

1.5386  240  1.6105  100 

6 

1.3022  6012 

1.3400  956 

1.4185  191 

1.5007  304 

1.5668  743 

1.6771  001  1.7715  610 

7 

1.3608  6183 

1.4071  004 

1.5036  303 

1.6057  815 

1.7138  243 

1.8280  391  1.9487  171 

8 

1.4221  0061 

1.4774  554 

1.5938  481 

1.7181  862 

1.8509  302 

1.9925  626  2.1435  888 

9 

1 . 4860  9514 

1.5513  282 

1.6894  790 

1.8384  592 

1.9990  046 

2.1718933  2.3579477 

10 

1.5529  6942 

1.6288  946 

1.7908  477 

1.9671  514 

2.1589  250 

2.3673  637  2.5937  425 

11 

1.6228  5305 

1.7103  394 

1.8982  986 

2.1048  520 

2.3316  390 

2.5804  264  2.8531  167 

12 

1.6958  8143 

1.7958  563 

2.0121  965 

2.2521  916 

2.5181  701 

2.8126  648  3.1384  284 

13 

1.7721  9610 

1 .8856  491 

2.1329  283 

2.4098  450 

2.7196  237 

3.0658  046  3.4522  712 

14 

1.8519  4492 

1.9799  316 

2.2609  040 

2.5785  342 

2.9371  936 

3.3417  270  3.7974  983 

15 

1.9352  8244 

2.0789  282 

2.3965  582 

2.7590  315 

3.1721  691 

3.6424825  4.1772482 

16 

2.0223  7015 

2.1828  746 

2.5403  517 

2.9521  638 

3.4259  426 

3.9703  059  4.5949  730 

17 

2.1133  7681 

2.2920  183 

2.6927  728 

3.1588  152 

3.7000  181 

4.3276  334  5.0544  703 

18 

2.2084  7877 

2.4066  192 

2.8543  392 

3.3799  323 

3.9960  195 

4.7171  204  5.5599  173 

19 

2.3078  6031 

2.5269  502 

3.0255  995 

3.6165  275 

4.3157  Oil 

5.1416  613  6.1159  390 

20 

2.4117  1402 

2.6532  977 

3.2071  355 

3.8696  845 

4.6609  571 

5.6044  108  6.7275  000 

21 

2.5202  4116 

2.7859  626 

3.3995  636 

4.1405  624 

5.0338  337 

6.1088  077  7.4002  499 

22 

2.6336  5201 

2.9252  607 

3.6035  374 

4.4304  017 

5.4365  404 

6.6586  004  8.1402  749 

23 

2.7521  6635 

3.0715  238 

3.8197  497 

4.7405  299 

5.8714  637 

7.2578  745  8.9543  024 

24 

2.8760  1383 

3.2250  999 

4.0489  346 

5.0723  670 

6.3411  807 

7.9110  832  9.8497  327 

25 

3.0054  3446 

3.3863  549 

4.2918  707 

5.4274  326 

6.8484  752 

8.6230  807  10.8347  059 

26 

3.1406  7901 

3.5556  727 

4.5493  830 

5.8073  529 

7.3963  532 

9.3991  579  11.9181  765 

27 

3.2820  0956 

3.7334  563 

4.8223  459 

6.2138  676 

7.9880  615 

10.2450  821  13.1099  942 

28 

3.4296  9999 

3.9201  291 

5.1116  867 

6.6488  384 

8.6271  064 

11.1671  395  14.4209  936 

29 

3.5840  3649 

4.1161  356 

5.4183  879 

7.1142  571 

9.3172  749 

12.1721  821  15.8630  930 

30 

3.7453  1813 

4.3219  424 

5.7434  912 

7.6122  550 

10.0626  569 

13.2676  785  17.4494  023 

31 

3.9138  5745 

4 . 5380  395 

6.0881  006 

8.1451  129 

10.8676  694 

14.4617  695  19.1943  425 

32 

4.0899  8104 

4.7649  415 

6.4533  867 

8.7152  708 

11.7370  830 

15.7633  288  21.1137  768 

33 

4.2740  3018 

5.0031  885 

6.8405  899 

9.3253  398 

12.6760  496 

17.1820  284  23.2251  544 

34 

4.4663  6154 

5.2533  480 

7.2510  253 

9.9781  135 

13.6901  336 

18.7284  109  25.5476  699 

35 

4.6673  4781 

5.5160  154 

7.6860  868 

10.6765  815 

14.7853  443 

20.4139  679  28.1024  369 

36 

4.8773  7846 

5.7918  161 

8.1472  520 

11.4239  422 

15.9681  718 

22.2512  250  30.9126  805 

37 

5.0968  6049 

6.0814  069 

8.6360  871 

12.2236  181 

17.2456  256 

24.2538  353  34.0039  486 

38 

5.3262  1921 

6.3854  773 

9.1542  524 

13.0792  714 

18.6252  756 

26.4366  805  37.4043  434 

39 

5.5658  9908 

6.7047  512 

9.7035  075 

13.9948  204 

20.1152  977 

28.8159  817  41.1447  778 

40 

5.8163  6454 

7.0399  887 

10.2857  179 

14.9744  578 

21.7245  215 

31.4094  200  45.2592  556 

41 

6.0781  0094 

7.3919  882 

10.9028  610 

16.0226  699 

23.4624  832 

34.2362  679  49.7851  811 

42 

6.3516  1548 

7.7615  876 

11.5570  327 

17.1442  568 

25.3394  819 

37.3175  320  54.7636  992 

43 

6.6374  3818 

8.1496  669 

12.2504  546 

18.3443  548 

27.3666  404 

40.6761  09S  60.2400  692 

44 

6.9361  2290 

8.5571  503 

12.9854  819 

19.6284  596 

29.5559  717 

44.3369  597  66.2640  761 

45 

7.2482  4843 

8.9850  078 

13.7646  108 

21.0024  518 

31.9204  494 

48.3272  S61  72.8904  837 

46 

7.5744  1961 

9.4342  582 

14.5904  875 

22.4726  234 

34.4740  S53 

52.6767  419  80.1795  321 

47 

7.9152  6849 

9.9059  711 

15.4659  167 

24.0457  070 

37.2320  122 

57.4176  486  SS.1974  853 

48 

8.2714  5557 

10.4012  697 

16.3938  717 

25.7289  065 

40.2105  731 

62.5S52  370  97.0172  33S 

49 

8.6436  7107 

10.9213  331 

17.3775  040 

27.5299  300 

43.4274  190 

6S.2179  0S3  106.71S9  572 

50 

9.0326  3627 

11.4673  998 

18.4201  543 

29.4570  251 

46.9016  125 

74.3575  201  117.390S  529 

INTEREST 


207 


REVIEW  PROBLEMS  IN  INTEREST 

661.  1.  Find  the  interest  of  $4723.69  for  2 yr.  8 mo.  14  da.  at  6%. 

2.  What  will  $528.50  amount  to  in  1 yr.  4 mo.  22  da.  at  4%? 

3.  Loaned  $4500  on  February  16,  1907,  at  5%.  Received  the  amount  due 
me  on  May  4,  1908 ; how  much  did  I receive  ? 

J.  What  sum  must  be  invested  at  4%  for  a child  10  years  old,  that  he  may 
receive  $5000  when  he  is  21  years  of  age? 

5.  What  principal  will  amount  to  $3273.40,  at  6%,  if  loaned  January  15, 
1906,  and  paid  August  3,  1908? 

6.  What  sum  of  money  will  amount  to  $3630  in  3 years  at  7 %? 

7.  What  principal  will  produce  $7  interest,  at  5%,  in  90  days? 

8.  What  sum  must  be  invested  in  a property  that  pays  7 \c/o  per  annum,  to 
produce  an  income  of  $300  a year? 

9.  What  sum  of  money  will  amount  to  $2562  in  9 months  at  9 % ? 

10.  At  what  rate  will  $120  gain  $21,  if  placed  on  interest  for  3 yr.  6mo? 

11.  If  $800  amounts  to  $832  in  180  days,  what  is  the  rate  per  annum  ? 

12.  In  what  time  will  $900  amount  to  $1005  at  5 per  cent,  per  annum? 

13.  In  what  time  will  $880  produce  $55  interest  at  5%  ? 

In  what  time  will  $500,  placed  on  interest  at  4%,  double  itself? 

15.  Jan.  1,  1908,  I borrowed  $1500  at  6%  ; in  what  time  will  I owe  $1770? 

16.  If  a man  buys  a bill  of  goods  amounting  to  $2762.48,  terms  60  days  or 
3%  off  for  cash,  how  much  does  he  save  by  borrowing  the  money  at  6%  and 
paying  cash  ? 

17.  A buys  a house  for  $6000,  paying  $2000  cash,  and  giving  a mortgage  at 
5%  for  the  balance.  At  the  end  of  two  years  he  finds  he  has  paid  $224  in  taxes 
and  $97.50  for  repairs,  and  has  received  $800  for  rent.  What  per  cent,  has  he 
gained  on  his  investment  of  $2000  (reckoning  the  house  to  be  still  worth  $6000)? 

18.  At  -what  rate  will  $2475,  loaned  April  6,  1907,  amount  to  $2559.94  on 
March  25,  1908  ? 

19.  A man  buys  a piano  for  $275  on  instalments,  paying  $35  down  and  $10 
a month.  The  price  of  the  piano  would  have  been  $250  cash.  What  rate  of 
interest  is  he  paying  ? 

20.  On  Jan.  13,  1908,  a man  borrowed  $5500,  at  5%,  with  which  he 
purchased  a piece  of  land.  He  afterwards  sold  the  land  for  $5950,  paid  the  loan, 
and  found  that  he  had  gained  $172.44.  On  what  date  did  he  sell? 

21.  April  1,  1904,  I borrowed  $10500  at  4J%  interest,  and  invested  it  in  a 
farm  at  $75  an  acre.  Aug.  15,  1906,  my  agent  sold  50  acres  at  $105  an  acre, 
charging  me  3%.  The  money  was  deposited  in  a bank  paying  2J%  on  deposits. 
April  1,  1908,  I sold  the  remainder  of  the  farm  at  $95  an  acre.  After  paying  the 
interest,  what  was  my  gain? 


208 


INTEREST 


22.  How  much  money,  invested  at  4%,  will  amount  to  $10000  in  8 yr. 
5 mo.  29  da.? 

23.  A firm  bought  goods  on  credit,  and  agreed  to  pay  7 % interest  on  each 
purchase  from  its  date.  Oct.  16,  1907,  goods  were  bought  to  the  amount  of  $268  j 
Dec.  31,  1907,  to  the  amount  of  $765.80  ; Feb.  29,  1908,  to  the  amount  of  $600  ; 
Apr.  1,  1908,  to  the  amount  of  $325.25.  If  full  settlement  was  made  Aug.  25, 
1908,  how  much  cash  was  paid  ? 

21^.  On  August  27,  1906,  Smith  sold  his  farm  for  $16000 ; the  terms  were, 
$4000  cash  on  delivery,  $5000  on  May  27,  1907,  $3000  on  Feb.  28,  1908,  and  the 
remainder  in  two  years  from  date  of  purchase,  with  6%  interest  on  all  deferred 
payments.  What  was  the  total  amount  paid  ? 

25.  Oct.  16,  1907,  goods  were  bought  to  the  amount  of  $268  ; Dec.  31,  1907, 
to  the  amount  of  $567.90  ; on  Feb.  27,  1908,  to  the  amount  of  $575  ; on  May  1, 
1908,  to  the  amount  of  $235.75.  The  firm  agreed  to  pay  7%  interest  on  each 
purchase  from  its  date.  If  full  settlement  was  made  Sept.  5,  1908,  how  much 
cash  was  paid  ? 

26.  How  much  money  must  I deposit  in  a savings  bank  during  each  of  four 
years,  in  order  to  be  able  to  draw  out  $440,  if  the  bank  pays  4%  interest  on 
its  deposits? 

27.  I bought  a house  and  lot  on  speculation  for  $14325;  4 months  23  days 
from  date  of  purchase,  I sold  the  property  for  $15840.  If  money  was  worth  74% 
per  annum,  how  much  more  did  the  transaction  yield  me  than  if  I had  lent  the 
purchase  money  at  interest? 

28.  An  attorney  collects  a claim  of  $875  with  ordinary  interest  thereon  from 
August  29, 1908,  to  December  14,  1908,  at  7 %.  If  the  attorney’s  rate  for  collect- 
ing is  10%,  what  net  proceeds  should  be  paid  to  the  creditor? 

29.  Find  the  difference  between  the  accurate  interest  and  the  common 
interest  on  $8750  from  May  12,  1908,  to  October  15,  1908,  at  6f  % per  annum. 

30.  A contractor  borrowed  a sum  of  money  for  3 months  24  days  at  6%. 
Not  having  sufficient  funds  to  meet  this  obligation  when  due,  he  paid  60%  of 
the  debt  and  accrued  interest  by  giving  his  check  for  $5054,  and  agreed  to  con- 
tinue the  balance  at  8%  per  annum  until  paid.  The  balance  was  paid  3 months 
24  days  later,  with  interest.  What  was  the  amount  of  the  second  pa}rmeut? 

31.  A man  engaged  in  business  was  making  12J%  annually  on  his  capital 
of  $16840.  He  quit  his  business  and  loaned  his  money  at  74%.  What  did  he 
lose  in  2 yr.  3 mo.  18  da.  by  the  change? 

32.  A tract  of  land  containing  516  acres  was  bought  at  $36  an  acre,  the 
money  being  loaned  at  5J%.  At  the  end  of  2 yr.  9 mo.  15  da.,  f of  the  land 
was  sold  at  $42  an  acre  and  the  money  loaned  at  5%  ; 1 yr.  2 mo.  IS  da.  later 
the  remainder  was  sold  at  $384  an  acre  and  the  money  loaned  at  6%.  Five 
years  after  borrowing  the  original  sum,  the  loans  were  collected  and  the  original 
sum  returned  with  interest.  What  was  gained? 


BANK  DISCOUNT 

662.  Bank  Discount  is  a deduction  made  from  the  face  of  a promissory 
note  or  draft  for  cashing  such  negotiable  paper  before  maturity.  This  deduction 
is  the  interest  of  the  sum  due  at  the  maturity  of  the  note  or  draft  for  the  number 
of  days  from  the  date  on  which  it  is  discounted  to  the  date  of  maturity. 

663.  It  is  called  bank  discount  because  one  of  the  chief  functions  of  a bank 
is  the  cashing,  or  buying,  of  such  commercial  paper.  When  such  notes  are 
cashed  by  individuals,  however,  the  same  rules  are  observed. 

664.  The  methods  of  bank  discount  depend  upon  State  laws  and  in  some 
cases  upon  local  customs.  The  methods  which  follow  comply  with  the  statute 
laws  of  Pennsylvania  and  the  practise  in  its  chief  money  center,  Philadelphia, 
and  banking  circles  contributory  to  it. 

665.  Since  finding  bank  discount  is  precisely  the  same  operation  as  finding 
interest,  such  method  of  operation  may  be  used  as  is  best  suited  to  the  particular 
problem  under  consideration.  In  Pennsylvania  the  legal  and  usual  rate  is  6%, 
and  as  the  term  rarely  exceeds  four  months,  some  modification  of  the  “ Day 
Method  ” is  usually  preferable. 

666.  Asa  calendar  may  not  always  be  at  hand  to  determine  upon  what  day 
of  the  week  the  day  of  maturity  may  fall,  we  give  a method  of  determining  that 
fact. 

What  is  the  date  of  maturity  of  a note  dated  Thursday,  September  3,  1908, 
drawn  for  three  months.  On  what  day  of  the  week  does  it  fall  ? 

Three  months  after  September  3,  1908,  is  December  3,  1908. 

27  days  in  September 
31  “ “ October 

30  “ “ November 

3 “ “ December 

91  days  in  the  term  to  maturity.  91-w=13  weeks. 

There  being  no  remainder,  the  date  of  maturity,  December  3,  1908,  falls  on 
the  same  day  of  the  week  as  the  day  of  the  date  from  which  reckoning  is 
made,  Thursday. 

667.  The  method  is  to  divide  the  number  of  days  from  the  date  of  reckon- 
ing, usually  the  date  of  the  note,  to  the  date  of  maturity,  by  7,  the  number  of  days 
in  a week  ; then  count  as  many  days  of  the  week  from  the  day  of  the  week  upon 
which  the  day  of  reckoning  falls  as  there  are  days  in  the  remainder ; this  will 
give  the  day  of  the  week  upon  which  the  date  of  maturity  falls.  If  this  be  Sat- 
urday or  Sunday,  the  true  date  of  maturity  will  be  the  next  business  day. 

668.  Days  of  grace  are  abolished  in  many  States,  but  where  recognized 
by  statute  the  date  of  maturity  falls  upon  the  last  day  of  grace. 

669.  It  is  not  common  in  Philadelphia  to  offer  interest-bearing  notes 
for  discount,  but  if  the  banks  accept  such  notes  for  discount,  the  discount 
is  reckoned  on  the  face  of  the  note ; country  banks  reckon  the  discount  on  the 


209 


210 


BANK  DISCOUNT 


amount  of  the  note,  i.  e.,  the  face  plus  the  interest  on  the  face  for  the  full  interest 
term  of  the  note. 

670.  The  due  date  of  a note  is  the  date  upon  which  the  maker  is  legally 
bound  to  make  payment  at  some  designated  bank  or  place  of  business,  and  is 
found  by  adding  to  the  date  of  the  note  the  time  expressed  in  it. 

671.  Should  the  due  date  of  a note  fall  upon  a Sunday,  a Saturday,  or  any 
legal  holiday,  the  note  is  due  and  payable  on  the  next  business  day  following, 
and  the  additional  day  or  days  must  be  added  to  the  term  of  discount. 

672.  The  term  of  discount  is  the  number  of  days  from  the  date  of  dis- 
count to  the  date  on  which  the  note  is  payable,  inclusive  of  both  days. 

673.  The  legal  holidays  in  the  State  of  Pennsylvania  are:  New  Year’s 
Day  (Jan.  1),  Lincoln’s  Birthday  (Feb.  12),  Washington’s  Birthday  (Feb.  22), 
Spring  Election  Day  (third  Tuesday  in  February),  Good  Friday,  Memorial  Day 
(May  30),  Independence  Day  (July  4),  Labor  Day  (first  Monday  in  September), 
General  Election  Day  ( Tuesday  after  the  first  Monday  in  November),  Thanks- 
giving Day  (by  custom  the  last  Thursday  in  November),  Christmas  Day  (Dec.  25). 
Should  any  of  these  legal  holidays  fall  on  Sunday,  they  are  observed  on  the 
Monday  following.  Saturday  is  a legal  holiday  in  Pennsylvania,  so  far  as  com- 
mercial paper  is  concerned,  and  paper  nominally  due  on  that  day  is  legally  due 
on  the  following  business  day. 

674.  The  face  of  a note,  less  the  interest  for  the  term  of  discount,  is  called 

the  proceeds. 

675.  To  find  the  proceeds  of  a note. 

Example  1. — A note  dated  Monday,  June  29,  190S,  at  90  days,  for  $1450  is 
discounted  July  1,  1908.  What  are  the  proceeds? 


/ ^ S O X .0  b X O 
3 b O 


/ O . O O 
2 /.  7 S 
/ 1/  2 tf.  2 S 


BANK  DISCOUNT 


211 


Explanation. — Operation  1. — To  find  on  what  date  the  90  days  will  expire,  or  the  nominal 
due  date. 

If  the  note  was  dated  June  29,  one  of  the  90  days  will  be  in  June,  leaving  89  days  ; 31 
days  will  be  in  July,  leaving  58  days  ; 31  days  will  be  in  August  leaving  27  days  to  be  in  September. 
Hence  the  90  days  will  end  September  27,  which  is  the  nominal  due  date. 

Operation  2 — To  find  on  what  day  of  the  week  the  nominal  due  date  will  fall,  or  to  find  the 
legal  due  date. 

Using  Monday,  June  29,  as  the  day  of  working  the  example,  we  find  1 day  left  in  June  ; 31 
days  in  July,  31  days  in  August  and  27  days  in  September  or  90  days  from  the  date  of  working  the 
example  to  the  due  date.  Dividing  90  by  7 we  get  12  weeks  and  6 days  ; 6 days  from  Monday  is 
Sunday.  Hence  September  27  is  Sunday,  and  the  legal  due  date  is  Monday,  September  28. 

Operation  3. — To  find  the  term  of  discount. 

This  note  was  discounted  July  1,  hence  the  bank  would  have  it  30  days  in  July  plus  1 day  for 
the  day  of  discount,  31  days  in  August  and  28  days  in  September,  or  90  days,  which  is  the  term  of 
discount. 

Operation  4. — Find  the  interest  on  $1450  at  6fo  for  90  days,  which  is  $21.75,  or  the  bank  dis- 
count, and  the  proceeds  are  the  difference  between  the  face  of  the  note  $1450  and  $21.75  the  discount, 
or  $1428.25. 

Example  2.- — A note  at  three  months,  dated  Monday,  May  1,  1908,  for 
$1290,  is  discounted  June  29,  1908.  Find  the  proceeds. 


Explanation. — Operation  1. — To  find  on  what  date  the  three  months  will  end. 

One  month  from  May  1 is  June  1,  two  mouths  is  July  1 and  three  months  is  August  1. 

Operation  2. — To  find  on  what  day  of  the  week  Aug.  1 will  fall. 

Using  Monday  June  29  as  the  day  of  working  the  example,  we  find  1 day  left  in  June  ; 31  days 
in  July  and  1 day  in  August  or  33  days  from  the  date  of  working  the  example  to  the  due  date. 
Dividing  33  by  7 we  get  4 weeks  and  5 days  ; 5 days  from  Monday  is  Saturday.  Hence  August  1 is 
Saturday  and  the  legal  due  date  is  Monday,  August  3. 

Operation  3. — To  find  the  term  of  discount. 

This  note  was  discounted  June  29,  hence  the  bank  would  have  it  1 day  in  June  plus  1 day  for 
the  day  of  discount,  31  days  in  July  and  3 in  August  or  36  days,  which  is  the  term  of  discount. 

Operation  4. — Find  the  interest  on  $1290  at  6%  for  36  days,  which  is  $7.74  or  the  bank  dis- 
count, and  the  proceeds  are  the  difference  between  the  face  of  the  note  $1290  and  $7.74,  the  discount,  or 
$1282.26. 


212 


BANK  DISCOUNT 


676.  Rule. — First  find  the  due  date  of  the  note  by  adding  to  the  date  of  the  note 
the  time  expressed  in  it;  if  the  resulting  date  falls  on  Saturday , Sunday,  or  a legal 
holiday,  extend  the  time  to  the  next  business  day.  When  the  legal  du.e  date  has  been 
determined,  find  the  interest  on  the  face  of  the  note  for  the  number  of  days  from  the 
date  of  discount  to  the  legal  due  date,  including  both. 

Note. — The  local  custom  observed  in  Philadelphia  and  some  other  cities,  of  including  both  the 
day  of  discount  and  the  day  of  maturity  in  the  term  of  discount,  gives  one  day  more  than  the  difference 
found  by  subtracting  the  dates.  As,  January  5 to  January  15=10  days  ; but  January  5 to  January  15, 
inclusive=ll  days. 


EXERCISE 

677.  Find  date  of  maturity  and  term  of  discount. 


Date  of  Note.  Day  of  Week. 

Time. 

Discounted.  When  Due. 

1.  Jan.  6,  ’08 

Monday 

60  da. 

Jan.  6,  ’08 

2.  Feb.  24,  ’08 

Monday 

2 mo. 

Mar.  9,  ’08 

3.  Dec.  26,  ’08 

Saturday 

2 mo. 

Jan.  15,  ’09 

J.  May  14,  ’08 

Thursday 

90  da. 

June  2,  ’OS 

5.  July  11,  ’08 

Saturday 

60  da. 

Aug.  6,  ’08 

6.  Oct.  14,  ’08 

Wednesday 

4 mo. 

Dec.  30,  ’08 

7.  May  19,  ’08 

Tuesday 

3 mo. 

June  18,  ’08 

8.  Jan.  7,  ’09 

Thursday 

90  da. 

Feb.  26,  ’09 

9.  Dec.  19,  ’08 

Saturday 

15  da. 

Dec.  19,  ’08 

10.  Nov.  25,  ’08 

Wednesday 

3 mo. 

Feb.  19,  ’09 

11.  Aug.  11,  ’08 

Tuesday 

2 mo. 

Sept.  7,  ’08 

12.  Sept.  19,  ’08 

Saturday 

30  da. 

Sept.  30,  ’08 

13.  May  4,  ’08 

Monday 

120  da. 

July  6,  ’08 

If.  Sept.  25,  ’08 

Friday 

4 mo. 

Nov.  2,  ’08 

15.  Jan.  15,  ’08 

Wednesday 

15  da. 

Jan.  24,  ’08 

16.  Mar.  5,  ’08 

Thursday 

4 mo. 

Apr.  18,  ’08 

17.  May  18,  ’08 

Monday 

2 mo. 

June  16,  ’08 

18.  Jan.  15,  ’08 

Wednesday 

90  da. 

Feb.  1,  ’08 

19.  July  11,  ’08 

Saturday 

4 mo. 

Aug.  18,  ’08 

20.  Mar.  19,  ’08 

Thursday 

6 mo. 

June  9,  ’OS 

21.  May  21,  ’08 

Thursday 

30  da. 

June  1,  ’08 

22.  Apr.  20,  ’08 

Monday 

15  da. 

Apr.  IS,  ’08 

23.  Dec.  18,  ’08 

Friday 

90  da. 

Jan.  2,  ’09 

2f.  Aug.  14,  ’08 

Friday 

4 mo. 

Nov.  3,  ’08 

25.  Dec.  31,  ’08 

Thursday 

2 mo. 

Jan.  18,  ’09 

EXERCISE 

678.  Find  the  bank  discount. 

Date  of  Note  and 
Day  of  Week. 

Face. 

Time. 

,,  When  Date  of 

a e'  Discounted.  Maturity. 

1.  May  18,  ’08,  Mon. 

$400.00 

60  da. 

6%  May  18, ’08 

2 Apr.  9,  ’08,  Thurs. 

$500.00 

90  da. 

6%  Apr.  10, ’OS 

3.  Oct.  20,  ’08,  Tues. 

$1000.00 

30  da. 

6%  Oct.  20, ’08 

If.  Dec.  21,  ’08,  Mon. 

$387.25 

2 mo. 

6%  Dec.  21, ’08 

Term  of 
Discount. 


Bank 

Discount. 


BANK  DISCOUNT 


213 


5. 

May  9,  ’08,  Sat. 

$487.37 

3 mo. 

6 % May  9,  ’08 

6. 

Aug.  13,  ’08,  Thurs. 

$587.00 

1 mo. 

6%  Aug.  13, ’08 

7. 

Nov.  5,  ’08,  Thurs. 

$1250.00 

90  da. 

6%  Dec.  5, ’08 

8. 

June  11,  ’08,  Thurs. 

$1187.50 

3 mo. 

6%  July  14, ’08 

9. 

July  15,  ’08,  Wed. 

$958.75 

4 mo. 

6%  Aug.  29, ’08 

10. 

Jan.  6,  ’08,  Mon. 

$546.27 

90  da. 

6%  Feb.  10, ’08 

11. 

Dec.  26,  ’08,  Sat. 

$787.37 

2 mo. 

6%  Dec.  26, ’08 

12. 

Nov.  25,  ’08,  Wed. 

$1358.68 

3 mo. 

6%  Dec.  12, ’08 

13. 

May  14,  ’08,  Thurs. 

$186.75 

90  da. 

6%  June  20,  ’08 

U.  Aug.  21,  ’08,  Fri. 

$50.76 

2 mo. 

6%  Sept,  9, ’08 

15. 

Dec.  28,  ’08,  Mon. 

$75.80 

4 mo. 

6 % Dec.  31, ’08 

16. 

Dec.  30,  ’08,  Wed. 

$75.86 

120  da. 

6%  Dec.  31, ’08 

17. 

Jan.  10,  ’08,  Fri. 

$47.50 

60  da. 

6%  Feb.  3,  ’08 

18. 

July  14,  ’08,  Tues. 

$57.67 

3 mo. 

6%  Aug.  10,  ’08 

19. 

Sept.  11,  ’08,  Fri. 

$463.87 

20  da. 

6%  Sept.  14,  ’08 

20. 

July  13,  ’08,  Mon. 

$50.27 

10  da. 

6%  July  13, ’08 

21.  Apr.  15,  ’08,  Wed. 

$408.75 

8 da. 

6%  Apr.  15,  ’08 

22. 

June  11,  ’08,  Thurs. 

$568.70 

18  da. 

6%  June  17, ’08 

EXERCISE 

679.  Find  the  proceeds. 

Date  of  Note  and 

Face. 

Time. 

When  Bank 

Day  of  Week. 

Discounted.  Discount. 

1. 

•Jan.  9,  ’08,  Thurs. 

$450.00 

90  da. 

•Jan.  9,  ’08 

2. 

May  14,  ’08,  Thurs. 

$750.00 

30  da. 

May  14,  ’08 

3. 

Mar.  13,  ’08,  Fri. 

$950.00 

60  da. 

iMar.  14,  ’08 

If..  Apr.  20,  ’08,  Mon. 

$868.00 

1 mo. 

May  9,  ’08 

5. 

Dec.  21,  ’08,  Mon. 

$567.27 

3 mo. 

Jan.  4,  ’09 

6. 

Jan.  20,  ’09,  Wed. 

$50.70 

10  da. 

Jan.  20,  ’09 

7. 

Nov.  19,  ’08,  Thurs. 

$487.70 

20  da. 

Nov.  28,  ’08 

8. 

Dec.  26,  ’08,  Sat. 

$460.75 

2 mo. 

Jan.  9,  ’09 

9. 

Dec.  26,  ’08,  Sat. 

$467.50 

60  da. 

Jan.  9,  ’09 

10. 

June  11,  ’08,  Thurs. 

$1275.00 

120  da. 

July  6,  ’08 

11. 

Oct.  20,  ’08,  Tues. 

$1100.00 

30  da. 

Nov.  2,  ’08 

12. 

Dec.  21,  ’08,  Mon. 

$50.00 

1 mo. 

Jan.  4,  ’09 

13.  Sept.  5,  ’08,  Sat. 

$10.25 

10  da. 

Sept.  5,  -0S 

Ilf.  July  15,  ’08,  Wed. 

$1375.87 

20  da. 

Aug.  1,  ’08 

15. 

May  9,  ’08,  Sat. 

$987.70 

3 mo. 

June  1,  ’08 

16. 

Dec.  21,  ’08,  Mon. 

$787.70 

90  da. 

Feb.  1,  ’09 

17. 

Sept.  21,  ’08,  Mon. 

$1500.00 

10  da. 

Sept.  22,  ’08 

18. 

Oct.  21,  ’08,  Wed. 

$1476.80 

15  da. 

Oct.  28,  ’OS 

19. 

Dec.  31,  ’08,  Thurs. 

$1156.76 

3 mo. 

Feb.  17,  ’09 

20. 

Jan.  4,  ’09,  Mon. 

$987.40 

60  da. 

Feb.  9,  ’09 

21. 

Feb.  20,  ’09,  Sat. 

$568.95 

4 mo. 

Mar.  20,  ’09 

Proceeds. 


214 


BANK  DISCOUNT 


MENTAL.  PROBLEMS 

680.  1.  What  is  the  bank  discount  of  a note  of  $60,  discounted  for  90  days, 
at  6 per  cent.? 

Solution. — For  60  days  the  discount  is  60  cents,  and  for  90  days,  which  is  a half  more  than  60 
days,  the  discount  is  a half  more  than  60  cents,  or  90  cents. 

2.  What  is  the  bank  discount  of  $200  for  GO  days? 

3.  What  is  the  bank  discount  of  $S0  for  60  days? 

J.  What  is  the  bank  discount  of  $120  for  30  days? 

5.  What  is  the  bank  discount  of  $200  for  120  days  ? 

What  is  the  bank  discount  of  $90  for  30  days? 

What  is  the  bank  discount  of  $180  for  60  days? 

What  is  the  bank  discount  of  $240  for  63  days? 

What  is  the  bank  discount  of  $600  for  120  days  ? 

What  is  the  bank  discount  of  $300  for  20  days  ? 

What  are  the  proceeds  of  a note  for  $800  discounted  for  30  days? 

What  are  the  proceeds  of  a note  for  $240  discounted  for  30  days? 

What  are  the  proceeds  of  a note  for  $400  discounted  for  90  days  ? 

What  are  the  proceeds  of  a note  for  $200  discounted  for  40  days  ? 

What  are  the  proceeds  of  a note  for  $600  discounted  for  15  days? 

What  are  the  proceeds  of  a note  for  $60  discounted  for  20  days  ? 

17.  What  are  the  proceeds  of  a note  for  $120  discounted  for  12  days? 

18.  What  are  the  proceeds  of  a note  for  $200  discounted  for  15  days? 

What  are  the  proceeds  of  a note  for  $300  discounted  for  36  days  ? 

What  are  the  proceeds  of  a note  for  $540  discounted  for  180  days? 

681.  To  find  the  face  of  a note. 

Example. — What  is  the  face  of  a note  at  90  days,  dated  June  29,  1908,  and 
discounted  on  that  date,  that  will  yield  $1200  proceeds? 

June  29+90  days=Sunday,  September  27,=Monday,  September  28. 
From  June  29  to  September  28,  inclusive=92  days. 

$1  * ,984f  ) 1200 

92  3 3 

1)2 


6. 

7. 

8. 
9. 

10. 

11. 

12. 

13. 

n. 

15. 

16. 


19. 

20. 


6PPJJ  ) 


2.954  ) 


•015* 


3600.0000  ( 
2954 
6460 
590S 


1218.68  face  of  note 


$1,000 


•015* 


$.984§  = Proceeds  of  $1 


5520 

2954 

25660 

23632 

20280 

17724 

25560 

23632 


19280 


Result  S121S.69. 


682.  Divide  the  given  proceeds  by  the  proceeds  of  $ 1 . 


BANK  DISCOUNT 

215 

EXERCISE 

883.  Find  the  face  of  note. 

Date  of  Note. 

Face.  Time. 

Date  of  Discount. 

Proceeds. 

1.  Jan.  10,  ’08,  Fri. 

60  da. 

Jan.  10,  ’08 

$450.00 

2.  Feb.  14,  ’08,  Fri. 

30  da. 

Feb.  14,  ’08 

$500.00 

3 Feb.  12,  ’08,  Wed. 

90  da. 

Feb.  12,  ’08 

$650.00 

If..  Mar.  10,  ’08,  Tues. 

2 mo. 

Mar.  10,  ’08 

$450.25 

5.  Apr.  16,  ’08,  Thurs. 

3 mo. 

Apr.  16,  ’08 

$375.60 

6.  June  9,  ’08,  Tues. 

4 mo. 

June  9,  ’08 

$2000.00 

7.  Jan.  27,  ’08,  Mon. 

1 mo. 

Jan.  27,  ’08 

$3000.00 

8.  Dec.  30,  ’08,  Wed. 

2 mo. 

Dec.  31,  ’08 

$400.75 

9.  Dec.  26,  ’08,  Sat. 

2 mo. 

Jan.  4,  ’09 

$786.50 

10.  Nov.  25,  ’08,  Wed. 

90  da. 

Nov.  25,  ’08 

$10000.00 

11.  July  7,  ’08,  Tues. 

4 mo. 

July  7,  ’08 

$100.00 

12.  Aug.  14,  ’08,  Fri. 

60  da. 

Aug.  15,  ’08 

$50.00 

13.  July  18,  ’08,  Sat. 

90  da. 

July  20,  ’08 

$10.00 

Ilf..  Nov.  17,  ’08,  Tues. 

4 mo. 

Nov.  17,  ’08 

$487.75 

15.  May  19,  ’08,  Tues. 

30  da. 

May  20,  ’08 

$568.46 

16.  Dec.  3,  ’08,  Thurs. 

4 mo. 

Dec.  5,  ’08 

$787.80 

17.  Oct.  10,  ’08,  Sat. 

30  da. 

Oct.  12,  ’08 

$1160.75 

18.  Jan.  10,  '08,  Fri. 

4 mo. 

Jan.  14,  ’08 

$1230.50 

19.  Aug.  31,  ’08,  Mon. 

30  da. 

Aug.  31,  ’08 

$5.00 

20.  July  20,  ’08,  Mon. 

60  da. 

July  20,  ’08 

$1250.00 

- 21.  Feb.  21,  ’08,  Fri. 

90  da. 

Feb.  21,  ’08 

$1178.80 

22.  Jan.  21,  ’08,  Tues. 

120  da. 

Jan.  23,  ’08 

$125.75 

23.  Oct.  23,  ’08,  Fri. 

3 mo. 

Oct.  29,  ’08 

$499.60 

2If.  Jan.  4,  ’09,  Mon. 

90  da. 

Jan.  5,  ’09 

$568.90 

25.  June  19,  ’08,  Fri. 

30  da. 

June  30,  ’08 

$5000.00 

WRITTEN  PROBLEMS 

684.  1.  Find  the  proceeds  of  a note  for  $1190,  at  90  days,  dated  and  dis- 
counted to-day. 

2.  I send  to  bank  two  notes ; the  first  dated  to-day,  at  three  months,  for 
$2200;  the  second  dated  to-day,  at  sixty  days,  for  $1190.25.  If  both  these  notes 
are  discounted  and  the  proceeds  placed  to  my  credit,  what  is  my  bank  balance? 

3.  Find  the  proceeds  of  a note  dated  and  discounted  to-day,  at  three 
months,  the  face  of  which  is  $975.62. 

J.  What  is  the  discount  of  a note  at  five  months,  dated  four  weeks  ago 
and  discounted  to  day,  if  the  face  of  the  note  is  $1500? 

5.  What  were  the  proceeds  of  a note  for  $1600,  dated  January  18,  1908, 
Saturday,  at  90  days,  which  was  discounted  February  17,  1908? 

6.  Sent  to  bank  John  Alpine’s  60-day  note,  dated  7 days  ago,  for  $1500, 
and  bad  it  discounted  to-day.  Find  the  proceeds. 


216 


BANK  DISCOUNT 


7.  Sold  John  Halter  925  yards  broadcloth  at  $2.16,  terms  ninety  days. 
Received  His  note  which  I immediately  had  discounted.  Find  the  discount. 

8.  Hiram  Hillegas  to-day  discounted  his  note  in  my  favor,  given  14  days 
ago,  for  $1190  at  four  months.  Find  the  proceeds. 

9.  Fourteen  days  ago  I sold  Abram  Levis  1280  yards  linen  at  18  cents, 
on  a credit  of  sixty  days.  I received  his  note  and  to-day  had  it  discounted. 
Find  the  proceeds. 

10.  A merchant  sends  to  bank  $320  cash  ; John  Evans’s  note  dated  seven 
days  ago,  at  90  days,  for  $350  ; Abram  Ivinkler’s  note  for  $1250  dated  seven  days 
ago,  at  two  months ; William  Witherspoon’s  note  for  $1500,  dated  to-day,  at  90 
days  ; and  John  Lardon’s  note  for  $390  at  2 months,  dated  to-day.  If  all  these 
notes  are  discounted,  find  the  total  amount  placed  to  the  merchant’s  credit. 

11.  I drew  a draft  on  A.  Bearer,  of  New  York  City,  dated  to-day,  at  four 
months,  for  $3000,  and  offered  it  for  discount  at  my  bank,  and  it  was  accepted. 
What  amount  should  be  placed  to  my  credit? 

12.  What  is  the  net  amount  of  a note,  dated  December  30,  1908,  Wednes- 
day, at  four  months,  for  $1200,  discounted  February  1,  1909? 

13.  What  amount  will  a sixty-day  sight  draft  for  $5000,  dated  June  5, 
1908,  Friday,  yield  if  discounted  June  10,  1908? 

7J.  I hold  A.  Buyer’s  90-day  note,  for  $1875,  dated  one  week  ago.  If  my 
bank  discounts  it  for  me  to-day,  what  sum  will  be  placed  to  my  credit? 

15.  I want  to  secure  $1000  cash  to-day,  and  to  do  so  I have  discounted  my 
note  at  90  days,  dated  to-dav.  Find  the  face  of  the  note. 

16.  I have  in  bank  a balance  of  $98.50.  Wishing  to  give  a check  for  $700, 

I have  discounted  my  note  dated  to-day,  at  three  months,  for  a sum  sufficient  to 
secure  the  balance.  Find  the  face  of  my  note. 

17.  John  Banderlin  has  in  bank  a balance  of  $19.20.  He  sends  to  bank 
$250  cash,  Warren  Baton’s  note  dated  to-day,  at  three  months,  for  $1120,  and 
his  own  note  at  ninety  days,  dated  to-day,  for  a sum  sufficient  to  enable  him  to 
write  a check  for  $4500  and  have  $1500  remaining  in  bank.  Find  the  face  of 
his  own  note. 

18.  For  what  amount  must  I write  a note  at  four  months  that  will  yield  me  ' 
$2211  if  dated  and  discounted  to-day? 

19.  For  what  amount  must  I write  a note,  dated  to-day,  which  I intend  to 
have  discounted  the  next  business  day,  the  time  of  the  note  being  three  months, 
to  yield  $900  ? 

20.  I have  in  bank  to-day  a balance  of  $750.22.  I owe  a debt  of  $1200, 
and  in  order  to  secure  the  balance  I give  a note  with  collateral,  at  three  months, 
dated  to-day.  What  must  be  the  face  of  the  note? 

21.  Find  the  face  of  a note  that  will  yield  $1720,  if  the  time  of  the  note  is 
five  months  and  it  is  dated  and  discounted  to-day. 

22.  The  proceeds  of  a 90-day  note,  dated  and  discounted  to-day,  are 
$1115.42.  Find  its  face. 


BANK  DISCOUNT 


217 


33.  I owe  John  Harley. $1800  to-day,  and  lie  agrees  to  take  in  payment  my 
note  made  payable  to  William  Kessler,  at  three  months.  For  what  amount  must 
I write  the  note? 

Slf..  I have  in  bank  to-day  a balance  of  $950.  I send  to  bank  four  notes : 
the  first  for  $1125,  dated  to-day,  at  90  days;  the  second  for  $1050,  dated  to-day, 
at  two  months;  the  third  for  $975,  dated  7 days  ago,  at  60  days;  and  the  fourth  is 
my  own  note  at  30  days,  dated  to-day,  for  such  an  amount  as  will  make  my  bank 
balance  $5000.  Find  the  face  of  my  own  note. 

35.  Raise  $1000  to-day,  January  5,  1909,  Tuesday,  by  having  Merchant’s 
note,  dated  December  19,  1908,  for  $375,  at  four  months,  discounted  and  by 
giving  your  own  90-day  note,  dated  to-day,  for  such  an  amount  that  its  proceeds 
will  make  up  the  sum  needed. 

36.  Borrow  at  bank  $1800  on  three  notes  at  one,  two,  and  three  months, 
respectively.  The  notes  are  all  to  be  dated  and  discounted  to-day.  For  what 
sums  shall  they  be  drawn  so  that  the  proceeds  of  the  notes  shall  be  equal  ? 

37.  My  bank  discounts  for  me  to-day  an  interest-bearing  note,  at  four 
months,  dated  a week  ago.  The  proceeds  are  $540  ; what  is  the  face  of  the  note  ? 

38.  For  what  amount  must  a note  at  four  months,  dated  to-day,  bearing 
interest  at  5%,  be  drawn  so  that,  discounted  to-day  at  6%,  the  proceeds  will  be 
$900  9 

39.  If  I buy  goods  for  $1200  cash  and  immediately  dispose  of  them  for 
$1300,  receiving  a note  at  four  months,  which  I have  discounted  at  once, 
receiving  the  proceeds ; what  is  my  per  cent,  of  gain  on  the  sale? 

30.  A man  has  a 60-day  note  discounted,  term  of  discount  61  days,  and 
receives  $12.20  less  than  the  face  of  the  note.  What  sum  does  he  receive? 

31.  I desire  to  raise  $4200  to-day.  I have  two  notes,  the  first  for  $2722.46, 
dated  10  days  ago,  at  90  days,  and  the  other  for  $1200,  dated  15  days  ago,  at  60 
days;  if  I have  these  notes  discounted,  what  additional  sum  must  I borrow  to 
make  the  required  amount? 

33.  I received  notice  Oct.  6,  1908,  that  my  account  was  overdrawn  to  the 
extent  of  $18.75.  To  adjust  the  matter  and  leave  me  a balance  in  bank,  I send 
to  bank  two  notes,  one  for  $451.75,  dated  Aug.  20,  1908,  Thursday,  at  four 
months,  and  the  other  for  $60,  dated  Sept.  3, 1908,  Wednesday,  at  90  days;  if ‘the 
proceeds  are  placed  to  my  credit,  what  amount  may  I check  out  and  still  leave 
my  balance  $35.57  ? 

33.  I sold  a merchant  958  pounds  of  sugar  at  5f  cents  per  pound ; 3852 
pounds  of  tobacco  at  $7.35  per  pound.  In  settlement  I take  his  note  at  60  days 
for  this  amount  and  have  the  note  discounted  immediately  at  5%;  what  did  I 
receive  for  the  goods? 


PARTIAL  PAYMENTS 


685.  A partial  payment  is  a payment  of  a part  of  a debt  and  its  accrued 
interest. 

686.  It  is  customary  to  acknowledge  the  partial  payment  of  a note  by 
writing  on  its  back  the  amount  of  payment  and  the  date.  These  payments  are 
sometimes  called  indorsements. 

687.  There  are  a number  of  rules  for  determining  the  amount  due  at  any 
time  on  a debt  on  which  partial  payments  have  been  made.  The  most  impor- 
tant are  the  United  States  Rule  and  the  Mercantile  Rule. 


UNITED  STATES  RULE 

688.  The  United  States  Rule  is  the  method  adopted  by  the  United  States 
Supreme  Court,  and  is  used  by  courts  in  general. 

Example. — On  a note  of  $2800  bearing  interest  at  5 %,  dated  July  8,  1907, 
the  following  payments  were  made : August  9, 1907,  $25  ; October  24.  1907,  815  ; 
Dec.  18,  1907,  $10;  Feb.  7,  1908,  $500;  April  4,  1908,  $1000;  June  3, 1908,  $300. 
What  is  due  July  8,  1908? 


Operation 


Face  of  note  July  8,  1907, 

$2800.00 

First  payment  Aug.  9,  1907, 

$25.00 

Interest  on  face  of  note  to  first  payment  (31  da. ), 

12.06 

12.94 

New  principal  Aug.  9,  1907, 

2787.06 

Second  payment  Oct.  24,  1907, 

15.00 

Interest  on  new  principal  to  Oct.  24,  1907  (75  da.), 

529.03 

•Third  payment  December  18,  1907, 

10.00 

Interest  on  same  principal  to  Dec.  18,  1907  (54  da.), 

20.90 

Fourth  payment  Feb.  7,  1908, 

500.00 

525.00 

Interest  on  same  principal  to  Feb.  7,  1908  (49  da.), 

18.97 

68.90 

456  10 

New  principal  Feb.  7,  1908, 

2330.96 

Fifth  payment  April  4,  1908, 

1000.00 

Interest  on  new  principal  to  April  4,  1908  (57  da. ), 

18.46 

931.54 

New  principal  April  4,  1908, 

1349.42 

Sixth  i^ayment  June  3,  1908, 

300.00 

Interest  on  new  principal  to  June  3,  1908  (59  da.), 

11.06 

283.94 

New  principal  June  3,  1908, 

1060.43 

Interest  to  July  8,  1908  (35  da. ), 

5.16 

Amount  due  July  8,  1908, 

$1065.64 

218 


PARTIAL  PAYMENTS 


219 


689.  United  States  Rule. — Find  the  interest  on  the  face  of  the  note  from  the 
date  of  the  note  to  the  time  of  the  first  payment  ( time  by  compound  subtraction).  From 
the  first  payment  substract  this  interest  and  reduce  the  face  of  the  note  by  the  balance. 
Find  the  interest  on  this  balance  for  the  time  from  the  first  payment  to  the  second  pay- 
ment. From  the  second  payment  subtract  this  interest  and  reduce  the  debt  by  the  bal- 
ance. Proceed  in  this  way  until  all  the  payments  have  been  disposed  of.  Then  add 
the  interest  from  the  last  payment  to  the  date  of  settlement. 

Note. — Should  any  interest  item  exceed  a payment,  simply  make  a note  of  the  payment  and 
accumulated  interest,  and  find  interest  on  same  amount  to  next  payment.  If  the  sum  of  both  pay- 
ments exceeds  both  interest  items,  reduce  the  face  of  the  note  by  the  difference  ; if  not,  continue  in  this 
way  until  the  sum  of  the  payments  does  exceed  the  sum  of  the  interest  items,  and  then  reduce  the  debt 
by  the  difference. 

Interest  must  never  be  taken  on  a sum  larger  than  that  on  which  the  preceding  interest  was  taken. 


MERCANTILE  RULE 

690.  The  Mercantile  Rule  is  the  method  commonly  used  by  merchants. 

Example. — On  a note  of  $2800  bearing  interest  at  5%,  dated  July  8,  1907, 
the  following  payments  were  made:  Aug.  10, 1907,  $25  ; Oct.  25,  1907,  $15  ; Dec. 
19,  1907,  $10;  Feb.  8,1908,  $500;  Apr.  4,  1908,  $1000;  June  3,  1908,  $300. 
What  is  due  July  8,  1908? 


Operation 


Face  of  note  July  8,  1907, 

$2800.00 

Interest  to  July  8,  1908, 

140.00 

Amount  due  July  8,  1908. 
First  payment  Aug.  10,  1907, 

$25.00 

$2940.00 

Interest  to  July  8,  1908  (333  da.), 

1.16 

Second  payment,  Oct.  25,  1907, 

15.00 

Interest  to  July  8,  1908  (257  da.), 

.54 

Third  payment,  Dec.  19,  1907, 

10.00 

Interest  to  July  8,  1908  (202  da.), 

.28 

Fourth  payment,  Feb.  8,  1908, 

500.00 

Interest  to  July  8,  1908  (151  da.), 

10.49 

Fifth  payment,  April  4,  1908, 

1000.00 

Interest  to  July  8,  1908  (95  da. ), 

13.19 

Sixth  payment,  June  3,  1908, 

300.00 

Interest  to  July  8,  1908  (35  da.), 

1.46 

1877.12 

Amount  due  July  8,  1908, 

$1062.88 

691.  Mercantile  Rule. — Find  the  interest  on 

the  face  of  the  note 

to  the  date 

of  settlement,  if  the  date  of  settlement  is  not  later  than  one  year  from  date.  Add  this 
interest  to  the  face  of  the  note.  Find  the  interest  on  each  payment  from  the  date  on 
which  it  was  made  to  the  date  of  settlement  ( exact  days).  Add  the  several  payments 
and  interest  items,  and  subtract  the  sum  from  the  amount  of  the  note. 

Note. — Should  the  date  of  settlement  extend  beyond  a year  from  the  date  of  the  note,  settlement 
of  the  note  must  be  made  at  the  end  of  every  year  from  date,  then  for  the  fraction  of  the  year,  if  any. 


220 


PARTIAL  PAYMENTS 


We  present  a problem  worked  out  by  both  methods  with  the  object  of  showing  that  for  large 
amounts  there  is  a considerable  difference  between  the  results  obtained  by  the  United  States  and  the 
Mercantile  Rules.  When  worked  by  the  Mercantile  Rule,  the  difference  is  in  favor  of  the  payer  of 
the  note.  It  will  be  seen  that  the  Mercantile  Rule  method  is  used  for  small  amounts  and  short  periods 
of  time. 


692.  $12000. 


Philadelphia.  January  3,  1906. 
On  demand  I promise  to  pay  John  Doe  or  order 


Twelve  thousand 

with  interest  at  6%,  without  defalcation  for  value  received. 


xxDollars, 

100 


Richard  Roe. 

Payments  indorsed:  March  3,  1906,  $2000;  May  3,  1906,  $2000;  July  3, 
1906,  $2000  ; Sept.  3, 1906,  $2000 ; Nov.  3, 1906,  $2000.  What  amount  will  take 
up  the  note  January  3,  1907  ? 


United  States  Rule 


Face  of  note  January  3,  1906, 

$12000.00 

Payment,  March  3,  1906, 

$2000.00 

Interest  on  face,  Jan.  3,  1906,  to  March  3,  1906, 

120.00 

1880.00 

Balance,  March  3,  1906, 

10120.00 

Payment,  May  3,  1906, 

2000.00 

Interest  on  balance,  March  3,  1906,  to  May  3,  1906, 

101.20 

1898.80 

Balance,  May  3,  1906, 

8221.20 

Payment,  July  3,  1906, 

2000.00 

Interest  on  balance,  May  3,  1906,  to  July  3,  1906, 

82.21 

1917.79 

Balance,  July  3,  1906, 

6303.41 

Payment,  Sept.  3,  1906, 

2000.00 

Interest  on  balance,  July  3,  1906,  to  Sept.  3,  1906, 

63.03 

1936.97 

Balance,  Sept.  3,  1906, 

4366.44 

Payment,  November  3,  1906, 

2000.00 

Interest  on  balance,  Sept.  3,  1906,  to  Nov.  3,  1906, 

43.66 

1956.34 

Balance,  November  3,  1906, 

2410.10 

Interest  on  balance,  Nov.  3,  1906,  to  Jan.  3,  1907, 

24.10 

Amount  to  take  up  note,  January  3,  1907, 

$2434.20 

Mercantile  Rule 

Face  of  note,  January  3,  1906, 

$12000.00 

Interest  for  one  year, 

720.00 

Amount  due  Jan.  3,  1907, 

12720.00 

Payment,  March  3,  1906, 

$2000.00 

Interest  on  payment  to  January  3,  1907,  306  da., 

102.00 

Payment,  May  3,  1906, 

2000.00 

Interest  on  payment  to  Jan.  3,  1907,  245  da., 

81.67 

Payment,  July  3,  1906, 

2000.00 

Interest  on  payment  to  Jan.  3,  1907,  184  da., 

61.33 

Payment,  Sept.  3,  1906, 

2000.00 

Interest  on  payment  to  Jan.  3,  1907,  122  da., 

40.66 

Payment,  November  3,  1906, 

2000.00 

Interest  on  payment  to  Jan.  3,  1907,  61  da., 

20.33 

10306.00 

$2414.00 

Amounts  needed  to  take  up  note  : 

By  United  States  Rule, 

$2434.20 

By  Mercantile  Rule, 

2414.00 

Difference  in  favor  of  payer  by  Mercantile  Rule, 

$20.20 

PARTIAL  PAYMENTS 


221 


WRITTEN  PROBLEMS 

693.  1.  A note  of  $1460,  dated  June  3,  1907,  with  interest  at  4J%,  has  the 
following  indorsements:  Aug.  6,  1907,  $150;  Nov.  19,  1907,  $250  ; Jan.  24,  1908, 
$210;  March  19,  1908,  $290;  May  20,  1908,  $100.  What  was  due  June  3,  1908, 
by  United  States  Rule? 

2.  On  a mortgage  of  $5000,  dated  March  6,  1907,  with  interest  at  5%,'  the 
following  payments  have  been  made:  May  23,  1907,  $950;  July  19,  1907,  $290; 
Oct.  17,  1907,  $450;  Dec.  18,  1907,  $1000;' Feb.  12,  1908,  $1500.  What  is  due 
March  6,  1908,  by  Mercantile  Rule? 

3.  A demand  note  dated  February  5,  1907,  bearing  interest  at  5J%,  for 
$2200,  has  the  following  indorsements  : May  15,  1907,  $25;  June  11, 1907,  $200  ; 
Aug.  20,  1907,  $10 ; Nov.  5,  1907,  $250  ; Jan.  9,  1908,  $300.  What  is  due  Feb.  5, 
1908,  by  United  States  and  Mercantile  Rules? 

J.  On  a note  dated  August  6,  1907,  for  $5200,  with  interest  at  6%,  the 
following  payments  have  been  made : Nov.  19,  1907,  $550 ; Jan.  2,  1908,  $30 ; 
March  27,  1908,  $33;  May  22,  1908,  $38;  July  23,  1908,  $42.  What  is  due 
Aug.  6,  1908,  by  United  States  Rule? 

5. 

$5000.  Philadelphia,  June  14,  1907. 

On  demand,  I promise  to  pay  to  the  order  of  Morris  and  Lewis, 
Five  thousand  Dollars  with  interest  at  5%,  without  defalcation,  value  received. 

H.  H.  Ettkr, 

104  S.  Eighteenth  St. 

On  the  above  note  the  following  indorsements  have  been  made : July  30, 

1907,  $100;  Sept.  16,  1907,  $10;  Nov.  18,  1907,  $25;  Feb.  11,  1908,  $1000; 
March  18,  1908,  $90;  May  13,  1908,  $25.  What  is  due  one  year  from  date  by 
United  States  and  Mercantile  Rules? 

6.  A note  of  $2500,  dated  Aug.  12,  1907,  with  interest  at  6%,  has  the 
following  indorsements : Oct.  26,  1907,  $400 ; Dec.  9,  1907,  $40 ; Jan.  24,  1908, 
$350;  May  25,  1908,  $25;  July  15,  1908,  $250;  Sept.  16,  1908,  $500;  Nov.  5, 

1908,  $200.  How  much  is  due  Jan.  1,  1909,  bv  United  States  Rule? 

7.  What  is  due  Mar.  7,  1908,  on  a note  for  $1878.90,  with  interest  at  6%, 
dated  Jan.  12,  1907,  on  which  the  following  payments  have  been  made:  Mar.  13, 
1907,  $100  ; May  18,  1907,  $200 ; June  10,  1907,  $75  ; Aug.  19,  1907,  $150  ; Oct. 
14,  1907,  $250  ? (Mercantile  Rule.) 

8.  What  sum  will  take  up  January  1,  1909,  a note  for  $10000,  interest  at 
4J%,  bearing  date  of  September  5,  1907,  upon  which  the  following  indorsements 
have  been  made:  January  2,  1908,  $2000;  April  1,  1908,  $2000;  July  1,  1908, 
$2000:  Oct.  2,  1908,  $2000? 


STOCKS  AND  BONDS 

694.  Stock  is  the  share  capital  of  a corporation  or  commercial  company ; 
that  is,  the  fund  employed  in  the  carrying  on  of  the  business. 

695.  The  capital  stock  of  a company  is  divided  into  shares  of  equal  amount, 
which  are  owned  by  the  individuals  who  jointly  form  the  corporation. 

696.  Certificates  of  stock  are  papers  issued  by  a company,  signed  by 
the  proper  officers,  indicating  the  number  of  shares  each  stockholder  is  entitled 
to,  and  are  an  evidence  of  ownership ; they  are  transferable  by  assignment  and 
may  be  bought  or  sold  like  any  other  property. 

697.  The  par  value  of  a share  of  stock  is  the  amount  named  in  the  certifi- 
cate— usually  $100,  though  it  may  be  any  amount,  as  $50,  $25,  $10,  etc.  Shares 
of  $50  are  sometimes  called  half-stock , and  those  of  $25,  quarter-stock. 

698.  The  market  value  of  a share  of  stock  is  the  price  at  which  it  is  sold. 

699.  When  the  market  value  of  shares  is  greater  than  the  par  value,  they 
are  said  to  be  above  par  or  at  a premium , and  when  the  market  value  is  less  than 
the  par  value,  they  are  said  to  be  below  par  or  at  a discount. 

700.  Quotations  of  the  market  value  are  generally  a percentage  of  the  par 
value,  though  they  sometimes  indicate  the  number  of  dollars  per  share. 

701.  The  gross  earnings  of  a company  are  the  total  receipts  before  expenses 
have  been  paid. 

702.  The  net  earnings  are  what  is  left  of  the  gross  earnings  after  paying 
expenses. 

703.  The  accumulated  profits  which  are  distributed  among  the  shareholders 
once  or  twice  a year  are  called  dividends,  and  are  “ declared  ” at  a certain  per 
cent,  of  the  par  value  of  their  shares. 

704.  An  assessment  is  a sum  levied  upon  stockholders  to  make  up  losses 
or  to  provide  for  extensive  improvements. 

705.  Preferred  stocks  are  shares  entitled  to  a dividend  of  a certain 
per  cent,  before  the  common  stock  can  receive  any  dividend.  Preferred  stock 
is  sometimes  issued  as  a special  inducement  to  raise  money  during  financial 
embarrassment. 

706.  The  term  watered  stock  is  applied  to  stock  which  has  been  increased 
by  shares  issued  in  excess  of  the  capital  stock  subscribed  for  and  actually  paid  in. 

707.  A bond  is  the  written  obligation  of  a Corporation,  City,  County,  State, 
or  Government,  to  pay  a certain  sum  of  mone}r  at  a certain  time  with  a fixed  rate 
of  interest,  payable  at  certain  periods.  The  bonds  of  business  corporations  are 
usually  secured  by  a mortgage  on  the  whole  or  some  specified  portion  of  their 
property. 

708.  Coupon  bonds  are  those  with  small  certificates  of  interest  attached, 
which  are  to  be  cut  off  and  presented  for  payment  as  they  become  due.  These 
bonds  and  coupons  are  signed  by  the  proper  officers,  and,  being  payable  to 
bearer,  are  transferable  by  delivery. 


222 


STOCKS  AND  BONDS 


223 


709.  Registered  bonds  are  those  payable  to  the  registered  owner  or  to  his 
order;  they  are  transferable  by  assignment.  The  interest  is  paid  by  check  or  in 
cash  to  the  owner. 

710.  Bonds  are  issued  in  various  denominations,  $1000  being  the  most 
common.  They  are  always  quoted  at  so  much  per  cent,  of  the  par  value. 

711.  The  Stock  Exchange  is  a place  where  stock  brokers  meet  daily  to 
execute  their  customers’  orders  for  the  purchase  and  sale  of  railroad  stocks  and 
bonds,  Government  securities,  and  such  other  stocks  and  bonds  of  various  cor- 
porations as  may  be  listed  at  the  Exchange. 

712.  The  regular  commission  for  buying  or  selling  at  the  New  York  Stock 
Exchange  is  one-eighth  of  one  per  cent,  on  the  par  value  of  all  stocks  and  bonds; 
that  is,  $12.50  on  100  shares  of  stock  of  the  par  value  of  $100  each,  or  $1.25 
on  a $1000  bond. 

At  the  Philadelphia  Stock  Exchange  the  rates  of  commission  areas  follows  : 

Rates  of  Commission 

PHILADELPHIA  STOCK  EXCHANGE 

United  States  Loans on  par  value,  1 per  cent. 

Other  Bonds  and  Loans on  par  value,  1 per  cent. 

Bank,  Insurance,  and  Trust  Company  Stocks  selling  at  or  -i 

over  one  hundred  dollars j ^ei  s^are>  cents. 

On  all  other  Shares,  selling  at  or  over  ten  dollars per  share,  12i  cents. 

Selling  under  ten  dollars per  share,  61  cents. 

The  charge  on  a single  share  varies  from  25  cents  to  $1. 

EXCEPTIONS  TO  THE  FOREGOIXG  COMMISSION  RULES 

Reading  com.  and  1st  and  2nd  pref.,  P.  R.  R.,  and  Del.,  Lack  a-  -i 

wanna  and  Western,  without  regard  to  selling  price  . . . . J Per  s^iare’  *h  cents- 

713.  A margin  is  a sum  of  money,  or  its  value  in  securities,  deposited  with 
a broker  to  protect  him  against  loss  in  buying  or  selling  for  a customer’s 
account.  It  is  usually  10%  of  the  par  value  of  the  stocks  bought  or  sold. 

Note. — Brokers  allow  their  customers  interest  on  money  deposited  as  margin  ; and  when  a cus- 
tomer buys  stock  ou  margin,  since  the  broker  has  to  pay  the  whole  cost  of  the  stock  in  cash,  he  charges 
his  customer  interest  on  the  amount  so  expended  for  his  account. 

714.  Stocks  are  usually  bought  or  sold  either  “regular,”  “cash,”  “seller 
three”  (s3),  or  “buyer  three”  (b3).  Stock  sold  in  the  regular  way  is  to  be 
delivered  the  next  day  ; if  sold  “ cash  ” it  is  to  be  delivered  on  the  day  sold. 
When  stock  is  bought  “seller  three  ” it  means  that  the  seller  can  deliver  it  on 
either  of  three  days  at  his  option — the  day  sold,  the  next  day,  or  the  day 
after  next.  “ Buyer  three”  means  that  the  buyer  may  demand  delivery  of  the 
stock  at  any  time  within  three  days,  but  must  take  and  pay  for  it  before  the 
end  of  the  third  day. 

When  stock  is  sold  at  either  buyer’s  or  seller’s  option  for  more  than  three 
days,  (as  blO,  b20,  b30,  b60 ; slO,  s20,  s30,  s60),  the  buyer  pays  six  per  cent, 
interest,  unless  “ flat”  is  specified  in  the  bargain. 


224 


STOCKS  AND  BONDS 


715.  Selling  short  means  selling  stock  which  one  does  not  own — either 
by  seller’s  option  (in  which  case  the  seller  buys  and  delivers  the  stock  before  the 
expiration  of  the  time  specified)  or  by  borrowing  the  stock  in  order  to  make 
delivery,  and  afterwards  buying  to  repay  the  loan.  In  either  case  the  person 
making  the  “short ’’sale  expects  to  be  able  to  buy  at  a lower  price,  and  soon 
enough,  to  make  a profit. 

710.  In  stock-exchange  slang,  a “ bull  ” is  one  who  endeavors  to  effect  arise 
in  the  price  of  stock,  and  a “ bear”  is  one  who  endeavors  to  bring  down  prices. 
A “ bull  ” buys  stock,  expecting  to  sell  it  at  a higher  price ; while  a “ bear  ” sells 
stock  “ short,”  expecting  to  buy  it  at  a lower  price. 

717.  A corner  is  a combination  to  buy  all  the  available  supply  of  a 
stock,  so  that  those  who  have  sold  short  may  be  unable  to  fulfil  their  contracts 
except  by  buying  of  the  combination  at  an  exorbitant  price. 

718.  To  hypothecate  stocks  or  bonds  is  to  deposit  them  as  collateral 
security  for  money  borrowed. 

STOCKS 

719.  To  find  cost  when  buying  or  proceeds  when  selling. 


Example  1. — What  must  I pay  for  150  shares 
C.  B.  & Q.  R.  R stock,  at  102J ; usual  brokerage  ? 

Quotation  1021  plus  brokerage  1 equals  102f. 


//  0 zf* 

/ ,6- O’ 
S.  / 0 o 
/ 0 2 


#3 .7, r 


S3  J3 . 


Example  2. — What  do  I obtain  from 
the  sale  of  150  shares  of  N.  Y.  C.  R.  R.  stock, 
at  118J  ; usual  brokerage? 

Quotation  118f  less  brokerage  1 equals  118|. 


/ 

/Jo 

Sq  o o 

/ / T 


-f-7/ 


/ 7 7 3 3 .7  S 


720.  R ule. — Add  the  brokerage  to  the  quotation  and  multiply  this  sum  ( expressed 
as  dollars ) by  the  number  of  shares  to  find  the  cost  of  purchase  ; subtract  the  brokerage 
from  the  quotation  and  multiply  this  difference  {expressed  as  dollars)  by  the  number  of 
shares  to  find  the  proceeds  of  sale. 

Note. — When  the  par  value  is  other  than  100,  multiply  by  quotation  expressed  as  dollars  and 
afterwards  add  or  subtract  brokerage,  as  the  case  may  require. 

A list  of  the  stocks  named  in  the  problems  of  this  book  whose  par  value  is  other  than  §100  per 
share  follows  : 

Catawissa  R.  R,  $50;  Lehigh  Valley  R.  R.,  $50  ; Lehigh  Navigation,  850; 
North  Penn.  R.  R.,  $50 ; Pennsylvania  R.  R.,  $50;  Reading  R.  R..  $50 ; P.,  11  il. 
& Balto.,  $50 ; Pitts.,  Gin.  & St.  L.,  $50. 


STOCKS  AND  BONDS 


225 


WRITTEN  PROBLEMS 

721.  1.  What  must  I pay  for  750  shares  Catawissa  preferred  at  53? 

Note. — When  no  brokerage  is  given,  the  usual  brokerage  of  1%,  or  121  cents  per  share,  par  $100, 
is  understood. 

S.  Through  my  broker  I sold  1250  shares  L.  V.  R.  R.  at  24J ; what  should 
I receive  from  him? 

3.  Bought  600  shares  Lehigh  Navigation  at  40,  and  sold  them  at  42,  usual 
brokerage  ; what  amount  of  money  do  I gain  ? 

4-  My  broker  buys  for  me  800  shares  P.  R.  R.  at  59,  and  350  shares  North 
Penn,  at  91 J ; what  is  the  full  cost  to  me? 

5.  What  will  it  cost  me  to  buy  250  shares  United  of  N.  J.  at  249J-? 

6.  I bought  500  shares  Un.  Gas  Imp.  at  87§,  and  to-day  it  is  quoted  at 
107f ; what  would  I gain  in  net  money  by  selling  at  once? 

7.  Bought  Pacific  Mail  at  57f,  and  sold  it  during  a decline  at  41  f ; what 
did  I lose  per  share  ? 

8.  Sold  750  shares  Welsbach  Light  at  38f  ; what  was  the  amount  of  my 
proceeds? 

9.  Plow  many  shares  of  Chi.  R.  I.  & P.  R.  R.  at  92J  can  I buy  with  $10000, 
and  what  surplus  will  remain? 

10.  I sold  350  shares  Lehigh  Nav.  at  47U  and  invested  in  P.  R.  R.  at  594; 
how  many  shares  did  I buy,  and  what  is  the  amount  of  surplus? 

11.  How  much  will  200  shares  of  Pennsylvania  Railroad  cost  at  53f,  if 
purchased  through  a Philadelphia  broker? 

IS.  How  much  will  200  shares  of  Pennsylvania  Railroad  cost,  if  purchased 
through  a New  York  broker  at  107? 

13.  How  many  shares  of  Reading  stock  at  llff-  can  I buy,  through  a Phila- 
delphia broker,  for  $2375? 

14-  What  is  the  cost  of  100  Reading  1st  preferred  at  23-ff,  200  N.  Pacific 
preferred  at  41f,  300  St.  Paul  at  87f,  and  200  Atchison  preferred  at  25§  ? 

15.  Find  the  cost  of  400  shares  Texas  & Pac.  at  10|,  brokerage  \%. 

16.  Find  the  proceeds  of  250  Omaha  preferred  at  145,  brokerage  \ 

17.  Purchased,  through  a New  York  broker,  100  Cent,  of  N.  J.  at  88J,  500 
Del.  & Hudson  at  1124,  300  Illinois  Cent,  at  99f,  and  200  Lake  Shore  at  171J. 
Find  cost. 

18.  Sold,  through  a Philadelphia  broker,  200  St.  Paul  at  86f,  150  Lehigh 
Yal.  at  30J,  425  Reading  at  Ilf,  and  300  West.  N.  Y.  & Pa.  at  2f.  Find  proceeds. 

19.  A broker  bought,  on  his  own  account,  500  St.  L.  & San  F.  1st  preferred 
at  47f  and  afterwards  sold  at  52J-;  what  was  his  gain? 

SO.  Find  the  proceeds  of  the  following  stocks  if  sold  through  a New  York 
broker:  100  Balt,  and  Ohio  at  114,  250  Wabash  preferred  at  15J,  200  Chi.  R.  I. 
& P.  R.  R.  at  76f , 150  West.  U.  Tel.  at  844,  25  Del.,  Lack.  & W.  at  158,  and  300 
Southern  Pac.  Co.  at  16|-. 


226 


STOCKS  AND  EONDS 


722.  Buying  and  selling  on  margin. 

Example. — On  June  4,  1908,  A.  Lamb  instructed  his  broker  to  buy  for  him 
200  shares  of  a certain  stock  and  deposited  with  him  $2000  margin.  On  June 
8,  he  bought  at  89§.  On  June  24  he  sold  the  stock  at  87J.  What  is  the  balance 
of  Lamb’s  account  July  2?  What  is  his  loss? 


Dr. 


A. 

A.  Lamb 


Cr. 


1908 

1908 

June 

8 

Bot.  200  shares  CL i. 

June  4 Cash  deposit 

2000  00 

R.  I.  & P.R.R.  at  89f 

17875  00 

“ 24  Sold  200  shares  Chi. 

U 

8 

Brokerage 

25  00 

R,  I.  & P.R.R,  at  87 4 

17500  00 

i ( 

24 

Brokerage 

25  00 

July  2 Int,  on  casli 

9 33 

July 

2 

Interest 

71  50 

“ 2 Int,  on  proceeds 

23  33- 

Interest 

10 

Interest 

03 

2 

Balance 

1536  03 

19532  66 

19532  66 

B. 

Dr.  A.  Lamb  Cr. 


1908 

1908 

June 

8 

Bot.  200  shares  Chi. 

June  4 Cash  deposit 

2000  00 

R.  I.  & P.R.R.  at  894 

17900 

00 

“ 8 Int.  on  bal. 

133 

“ 

24 

Int.  on  bal. 

42 

40 

“ 24  Sold  200  shares  Chi. 

July 

2 

Balance 

1536 

03 

R.  I.  & P.R.R,  at  87f 

17475  00 

July  2 Int.  on  bal. 

210 

19478 

43 

1947813 

Margin  deposited  $2000.00 

Balance  received  1536.03 


Loss  $463.97 

Notb. — Interest  is  charged  on  the  cost  of  stock  carried  and  is  allowed  on  each  deposit. 

Form  A.  is  given  as  showing  more  in  detail  the  charges  and  credits  of  a transaction  in  stocks 
bought  through  a broker  on  “margin.”  A.  Lamb  deposits  cash  to  the  amount  of  10  % of  the  par  value  of 
the  shares  he  instructs  his  broker  to  buy.  On  this  “ margin  ” Lamb  is  entitled  to  interest  until  the 
time  of  settlement,  in  this  case,  from  June  4,  1908,  to  July  2,  1908.  The  broker  advances  the  money  to 
buy,  upon  which  he  is  entitled  to  interest  until  time  of  settlement,  from  June  8,  1908,  to  July  2.  1908. 
He  charges  interest  on  brokerage  until  time  of  settlement.  From  June  24,  1908,  to  July  2,  1908,  the 
proceeds  are  in  the  hands  of  the  broker  and  he  must  pay  interest  to  Lamb  until  settlement,  from  June 
24,  1908,  to  July  2,  1908.  The  brokerage  is  shown  both  on  the  purchase  and  on  the  sale.  The  account 
shows  the  balance  due  Lamb  to  be  81536.03,  a loss  of  $463.97. 

Form  B.  shows  account  as  commonly  kept.  The  brokerage  being  added  to  purchase  and  deducted 
from  sale,  interest  being  calculated  on  daily  balances.  There  is  a credit  balance  of  $2000  from  June 
4 to  June  8,  on  which  the  interest  is  $1.33  ; a debit  balance  of  $15900  from  June  8 to  June  24.  on  which 
the  interest  is  $42.40,  and  a credit  balance  of  $1575  from  June  24  to  July  2,  on  which  the  interest  is 
$2.10. 


STOCKS  AND  BONDS 


227 


WRITTEN  PROBLEMS 

723.  1.  A speculator  directed  his  broker  to  purchase  500  shares  of  stock, 
par  $50,  and  deposited  10%  margin  on  April  6, 1908.  On  April  8 the  stock  was 
purchased  at  52J.  On  May  1 a dividend  of  3%  was  collected.  On  May  12 
the  stock  was  sold  at  54.  How  much  did  the  speculator  gain,  and  what  was  the 
balance  due  him  by  the  broker  on  May  14  ? 

2.  October  5,  1908,  my  broker  buys  for  me  500  shares  of  L.  V.  R.  R.  at 
29|,  brokerage  and  holds  it  till  February  25,  1909,  then  sells  it  at  34J, 
brokerage  as  before.  There  were  no  dividends  on  the  stock.  I put  up  $2500  as 
a margin  in  the  hands  of  the  broker  at  the  time  of  purchase.  What  does  he 
owe  me  at  the  time  of  sale  ? 

3.  Bought  150  shares  of  P.  R.  R.  at  55f,  received  a semiannual  dividend 
of  2J  per  cent,  and  then  sold  the  stock  for  55J.  How  much  did  I gain  ? 

4~  On  April  3,  1908,  a speculator  deposited  $1000  with  his  broker,  who 
purchased  for  him  100  shares  of  stock  at  89,  charging  \ % brokerage.  On  May  1, 
1908,  he  received  a dividend  of  $175.  On  October  14, 1908,  he  sold  the  stock  at 
92,  charging  |%  brokerage,  and  made  settlement,  charging  6%  interest.  How 
much  did  the  speculator  receive  ? 

724.  Buying  and  selling  short. 

Example. — Aaron  Bear  sold  “ short  ” December  1, 1908,  through  his  broker, 
250  shares  Northern  Pacific  R.  R.  pref.  at  60§,  and  “ covered  ” his  “short” 
December-15,  1908,  at  56.  Allowing  \ % brokerage  for  buying  and  selling,  what 
was  his  net  profit? 


Dr.  A.  Bear  Or. 


1908 

— 

1908 

Dec. 

1 

Commission 

31 

25 

Dec. 

1 

Deposit 

2500 

00 

U 

15 

Bought  250  N.  P. 

(< 

1 

Sold  250  N.  P.  pref. 

pref.  at  56 

14000 

00 

at  60| 

15093 

75 

u 

15 

Commission 

31 

25 

(C 

15 

Interest 

5 

83 

Ba  lance 

3537 

08 

17599,  58 

17599  58 

Balance  $3537.08 

Deposit  2500.00 

Gain  $1037.08 


228 


STOCKS  AND  BONDS 


Dr.  A.  Bear.  Cr. 


1908  ~ 

1908 

Dec. 

15 

Bot.  250  N.  P.  pref. 

Dec. 

1 

Cash  deposit 

2500  00 

at  56j 

14031 

25 

U 

1 

Sold  250  N.  P.  pref. 

(C 

15 

Balance 

3537 

08 

at  604 

15062  50 

u 

15 

Int.  on  cash 

5 83 

,17568 

33J 

17568  33 

Balance  $3537.08 

Margin  2500.00 

Gain  $1037.08 


Note. — On  “short”  sales  the  stock  is  borrowed  for  delivery  and  replaced  when  purchase  is 
made.  No  interest  is  charged  for  this.  The  margin  is  allowed  interest  as  in  any  other  stock  trans- 
action. 

PROBLEM 

A broker  sold  “short ’’for  I.  Flyer,  February  1,1908,700  shares  Illinois 
Steel  at  55  and  “covered”  sale  February  14,  1908,  at  52f.  What  was  his  net 
profit  ? 


BONDS 


725.  To  find  cost  when  bought  or  proceeds  when  sold. 

Example  1. — What  must  I pay  for  $12000,  par  value,  of  U.  S.  4s  regis- 
tered, at  113J  ? 

1134  + 1=1131 
$12000 
1-13| 

1356U00 

7500 


$13635  00  cost 

Example  2. — What  sum  should  my  broker  return  to  me  from  the  sale 
of  $8500,  par  value,  of  U.  S.  4s  coupon  at  114f  ? 

114f_ i=114| 

$8500 

1-14| 

34000 
93500 
_ 5313 

$9743.13  proceeds 

726.  Rule. — Add  rate  of  brokerage  to  quotation  and  multiply  it  by  tie  par 
value  of  the  bonds  to  find  cost;  subtract  rate  of  brokerage  from  quotation  and  multiply 
it  by  the  par  value  of  the  bonds  to  find  proceeds. 


STOCKS  AND  RONDS 


229 


WRITTEN  PROBLEMS 

727.  1 ■ What  will  be  the  cost  to  me  of  $12000,  par  value,  of  U.  S.  4s  coupon, 
at 128f ? 

2.  What  must  I pay  for  $8500  U.  S.  4s  registered,  at  127|? 

3.  Find  the  proceeds  of  $7000  U.  S.  os  coupon,  at  113§. 

4-  I sell  $5500  U.  S.  5s  registered  at  1134  and  invest  proceeds  in  U.  S. 
Currency  6s  at  103.  What  is  the  par  value  of  bonds  bought,  to  the  nearest 
$100,  and  what  the  surplus  ? 

5.  AVhat  will  $8000  Lehigh  Coal  & Nav.  5s  cost  at  96  (Philadelphia  rate 
of  brokerage)  ? 

6.  Bought  $15000  extended  2s  at  99J  and  sold  them  at  once  at  102 ; what 
was  my  gain  ? 

728.  To  find  rate  of  income. 

Example.— Bought  U.  S.  4s  registered,  at  113.  What  is  rate  of  income? 

4%  on  1134% 

1.131  ) 4.00  ( 

8 8 

9.05  ) 32.00  ( 3.53 -f  — 3.54%  nearly. 

27  15 

4 850 
4 525 

3250 

2715 

535 

729.  Rule  . — Divide  the  interest  rate  by  the  cost  ( quotation  plus  brokerage). 

WRITTEN  PROBLEMS 

730.  1.  AVhat  is  my  rate  of  income  from  U.  S.  new  4s  registered  bought  at 
1284? 

2.  AVhat  is  my  rate  of  income  from  City  of  Cincinnati  7^-s  bought  at  127|? 

3.  AVhat  would  be  my  rate  of  income  from  P.  & R.  Gen.  Mort.  4s  at  86f  ? 

4-  AVhat  net  rate  do  Catawissa  R.  R.  6s  pay  an  investor  if  bought  at  105f  ? 

5.  Which  is  the  better  investment — City  of  Camden  7s  at  114,  or  City  of 
Harrisburg  6s  at  112|? 

731.  To  find  the  cost  of  an  interest-bearing  bond  that  will  yield  a 
given  rate  of  income. 

Example. — At  what  price  must  I buy  5%  bonds  to  get  7 % on  my  invest- 
ment? No  brokerage. 

As  quotations  run  in  eighths,  it  is  evident  that 
the  result  obtained  must  be  modified  to  nearest  1 ) 5.00 

eighth,  which  would  be  a business  result,  but  not  743.  cosj-,.  Qr,  71  f business  result, 

a mathematical  one. 


230 


STOCKS  AND  BONDS 


732.  R ule. — Divide  the  rate  expressed  in  bond  by  the  rate  of  income  to  be 
obtained  and  the  result  will  be  the  cost. 

Note. — This  is  an  application  of  the  percentage  formula  : Percentage  divided  by  rate  gives  base. 


WRITTEN  PROBLEMS 


733.  1.  What  would  be  the  cost  of  bank  stock  paying  annual  dividends  of 
10%,  so  that  the  holder  would  realize  6%  on  his  investment?  No  brokerage. 

2.  A bank  stock  whose  par  is  $50  sells  at  140 ; semiannual  dividends  of 
10%  each  are  declared  each  year.  At  what  price  should  it  be  sold  to  make  it 
an  8 % investment.  No  brokerage. 

3.  At  what  price  must  I buy  U.  S.  new  4s  to  obtain  3%  on  my  investment? 
No  brokerage. 

Jf..  Southwark  Bank  stock  is  $50  per  share  at  par.  It  pays  20%  dividends. 
What  price  may  be  paid  for  it  per  share  to  realize  8%  on  the  cost? 

5.  At  what  price  must  I buy  U.  S.  Currenc\r  6s  to  make  5%  on  my  invest- 
ment ? 


734.  To  find  average  rate  of  income  derived  from  a bond  bought 
and  held  until  redeemed. 

Example. — "What  will  be  the  average  rate  of  income  from  a 4%  bond  bought 
at  118,  maturing  in  30  years?  No  brokerage. 


4%  for  30  years  120% 

Price  rec’d  on  redemp’n  100% 

Full  return  220% 

Cost  118 

Received  above  cost  102 

Average  annual  income 


1_0_2 
3 0 


yj -4-118  = 2.88 


3540  ) 102.00  ( 2.88 
70  80 

31  200 
28  320 

2 8800 


r concretely, 

Income  on  $1000  bond 

30  yr.  at  4 % 

$1200 

Bond  redeemed 

1000 

Full  return 

2200 

Cost 

1180 

Received  above  cost 

1020 

Average  annual  income 

34 

34^1180  = 2.88  or  2^ 

% nearly 

Result  2.88  approximately. 


735.  Rule.-- -Add  the  income  derived  from  the  bond  and  the  redemption  value, 
subtract  the  cost  from  this  sum  and  divide  the  difference  by  the  term  of  years ; this  mil 
give  the  average  annual  income  ; the  average  annual  income  divided  by  cost  will  give 
the  average  rate  of  income. 


STOCKS  AND  BONDS 


231 


736.  To  find  price  that  may  be  paid  for  a bond  drawing  a certain 
rate  of  interest  to  produce  a certain  rate  of  income. 


Example. — What  must  be  paid  for  a 4%  bond  maturing  in  30  years  to 
produce  3%  income? 


90  120 
100  100 
190  ) 220  ( 115.78 
190 
300 
190 
1100 
950 
1500 
1330 
1700 
1520 


Or  concretely, 

$1000  bond  at  4%  in  30  years 
will  produce  $1200 

Redemption  value  1000 

Result  115.78,  Full  return  2200 

approximately ; A3  % income  on  $1000 
or  1154,  nearly.  bond  would  give  in 

30  years  $900 

Redemption  value  1000 
Desired  return  1900 
1900  ) 2200.00  ( 115.78,  or  115-f  nearly. 


Remake:. — There  are  various  methods  in  use  for  determining  the  average  annual  income  in  cases 
of  this  kind,  some  being  based  upon  the  condition  that  the  annual  income  might  be  at  once  re-invested 
(compound  interest)  ; others  that  the  income  is  put  to  other  uses  than  investment.  The  above  method 
is  simply  a business  method  of  getting  a closely  approximate  result. 

737.  Rule. — Divide  full  return  received  from  bond , income  and  redemption,  by 
full  return  acceptable,  and  the  result  will  be  the  approximate  cost. 


PROBLEMS  IN  STOCKS  AND  BONDS 

738.  1.  Ho  w much  must  be  invested  in  Lehigh  Valley  6s  at  103J  to  produce 
a yearly  income  of  $1200? 

2.  What  quarterly  income  will  $29345.63  yield,  if  invested  in  U.  S.  new  4s 
coupon  at  124f,  usual  brokerage? 

3.  On  July  2,  I bought  300  Lehigh  Valley  at  28J,  b30,  and  called  for  the 
stock  July  21.  What  was  the  cost,  including  interest?  What  was  my  profit  if  I 
sold  on  July  21  at  31f  ? 

J.  What  was  paid  for  400  Lehigh  Nav.,  bought  July  6 at  41f,  s20,  and 
delivered  July  22? 

5.  How  much  invested  in  U.  S.  new  5s  registered  at  1144,  will  yield  an 
annual  income  of  $1500? 

6.  A man  having  $45000  to  invest,  bought  $10000  N.  Y.  P.  & N.  6s  at  104, 
$15000  Lehigh  Nav.  cons.  7s  at  129f,  and  $12000  S.  H.  & W.  1st  5s  at  106,  and 
deposited  the  balance  with  a trust  company  which  paid  2%  interest  on  deposits. 
Find  his  annual  income.  What  rate  per  cent,  is  it  on  his  investment? 

7.  Which  is  the  better  investment,  stock  paying  4J%  annual  dividends 
bought  at  75,  or  stock  paying  7 % dividends  bought  at  120? 

8.  What  per  cent,  income  on  the  investment  do  6 per  cent,  bonds  yield, 
if  purchased  at  104f  (brokerage  J%)? 


232 


STOCKS  AN  13  BONDS 


9.  Which  will  produce  the  greater  annual  income,  and  how  much,  $35000 
invested  instock  at  297  paying  5%  quarterly  dividends,  or  the  same  amount 
invested  in  stock  at  118  paying  3%  semiannually? 

10.  What  did  I pay  for  6%  stock  that  yields  me  4J%  on  my  investment? 

11.  Bought  through  a broker  sufficient  4%  bonds  to  produce  an  annual 
income  of  $1000.  What  did  they  cost  at  123§  and  what  is  the  per  cent,  of  income 
on  the  investment  ? 

12.  A invested  $25000  as  follows : 100  shares  of  5%  stock  at  102,  50  shares  of 
6%  stock  at  136f,  25  shares  of  3%  stock  at  80,  brokerage  |%,  and  loaned  the 
balance  on  mortgage  at  5%.  What  is  his  total  income? 

13.  How  much  invested  in  5%  bonds  at  122J  will  produce  an  annual 
income  of  $1600  ? 

Ilf..  At  what  price  must  4%  bonds  be  purchased  to  yield  3|%  on  the 
investment? 

15.  Which  yields  the  better  income  on  the  investment,  and  how  much  per 
cent.,  4%  stock  bought  at  121 J,  or  3%  stock  bought  at  92J? 

16.  What  is  paid  for  4-|-  % bonds  if  3|  % is  realized  from  the  investment  ? 

17.  If  I pay  124|  for  a 4 %bond  maturing  in  20  years,  what  per  cent,  income 
do  I receive  on  my  investment? 

18.  June  2,  1908,  a broker  purchased  for  the  account  of  a customer  600 
shares  of  stock  at  62|,  the  customer  depositing  $3000  margin.  July  8,  the 
stock  was  sold  at  71.  What  was  the  gain?  (Usual  commissions  and  interest.) 

19.  My  broker  sold  “short”  for  me  on  July  3,  1908,  500  shares  of  stock  at 
128|,  and  “ covered  ” on  July  22  at  126.  What  was  my  gain  ? 

20.  What  per  cent,  on  the  investment  do  6%  bonds  pay,  maturing  in  10 
years,  if  bought  at  112§-? 

21.  Stock  bought  through  a broker  at  79f  yields  4%  on  the  investment. 
What  per  cent,  dividend  does  the  stock  pay? 

22.  What  per  cent,  on  the  investment  do  5 per  cent,  bonds  pay,  maturing 
in  20  years,  if  bought  at  104f  ? 

23.  What  per  cent,  is  realized  on  the  investment  by  buying  8%  stock  at 
122? 

21^.  Smith  purchased,  through  a Philadelphia  broker,  500  Reading  at  12, 
200  Lake  Shore  at  170§,  and  100  Illinois  Cent,  at  99^.  He  sold  the  Reading  at 
12|,  the  Lake  Shore  at  172,  and  the  Illinois  Cent,  at  98f,  investing  the  proceeds 
in  U.  S.  4s  at  124J.  What  per  cent,  annual  income  does  he  realize  on  his  original 
investment  ? 

25.  How  much  must  be  paid  for  5%  bonds  maturing  in  12  years,  in  order  to 
realize  44%  on  the  investment  (allowing  §%  brokerage)? 

26.  How  much  must  be  paid  for  7 % bonds  to  realize  3%  on  the  investment 
if  the  bonds  mature  in  6 years  ? No  brokerage. 

27.  If  stock  bought  at  10%  premium  will  pay  6%  on  the  investment,  what 
per  cent,  will  it  pay  if  bought  at  20%  discount? 


STOCKS  AND  BONDS 


233 


28.  At  what  rate  should  stock  paying  annual  dividends  of  15%  be  bought 
to  realize  5J%  on  the  investment? 

29.  Chemical  Bank  Stock  is  quoted  at  4000 ; what  rate  of  dividend  should 
it  pay  to  give  the  investor  a 6%  income  ? 

30.  A man  buys  Bank  of  North  America  at  260,  which  pays  20% 
dividend  (par  $100).  What  is  the  rate  of  income  to  the  investor  ? No 
brokerage. 

31.  Bought  150  shares  of  P.  R.  R.  stock  at  55f,  received  a semiannual  divi- 
dend of  2J%,  and  then  sold  the  stock  at  55 \.  How  much  did  I gain  ? 

32.  Which  pays  the  better  interest,  railway  stock  at  147  (par  $50),  which 
pays  20%  dividends,  or  bank  stock  at  237  (par  $100),  which  pays  16%  dividends? 
No  brokerage. 

33.  A market  company  has  a capital  stock  of  $100000  divided  into  shares 
of  $100  each.  They  have  a bonded  debt  of  $250000.  Their  receipts  for  the  year 
are  $50000.  Expenses,  exclusive  of  interest,  $11000.  They,  of  right,  will  pay 
the  interest  on  their  bonded  debt  and  divide  the  balance  among  the  stockholders. 
What  rate  of  dividend  will  they  pay  the  stockholders? 

Slf..  Which  pays  the  better,  to  buy  a New  Jersey  7%  mortgage  at  5% 
discount,  or  a Pennsylvania  6%  mortgage  at  10%  discount?  No  brokerage. 

35.  What  is  my  rate  of  income  from  Reading  Terminal  5s  bought  at  118J? 
Usual  brokerage. 

36.  If  I buy  $10000  (par  value)  U.  S.  Gov.  4%  bonds  at  112,  payable  in  ten 
years,  and  receive  $400  each  year,  and  the  principal,  $10000,  at  the  end  of  the 
tenth  year,  wThat  rate  of  interest  have  I received  on  my  investment  ? 

37.  I bought  stock,  par  $50,  at  6J%  discount,  and  received  two  semiannual 
dividends  of  2J%  each  ; I then  sold  the  stock  at  7 % discount,  netting  a gain  of 
$50,  after  paying  a broker  \ % f°r  buying,  and  the  same  rate  for  selling.  How 
many  shares  were  bought? 

38.  A owned  sufficient  U.  S.  4s,  1907,  to  pay  him  an  annual  income  of  $300  ; 
be  sold  them  when  they  were  quoted  at  113R  brokerage  \°/o  and  invested  the 
proceeds  in  5%  R.  R.  stock  at  10%  premium,  brokerage  \ °/0  ; find  his  change  of 
income,  and  the  surplus  from  the  sale  of  the  bonds. 

39.  If  stock  bought  at  110  pays  411q%  on  the  investment,  what  rate  of  divi- 
dend does  the  stock  pay,  and  what  rate  of  interest  would  it  yield  if  the  price 
should  advance  20%  (brokerage  not  considered)? 

IpO.  What  rate  of  interest  does  an  investor  receive  on  his  money  who  buys 
through  a broker,  P.  R.  R.  stock,  par  value  $50,  at  $61,  if  this  stock  pays  a semi- 
annual dividend  of  2 J % , and  the  rate  of  brokerage  is  J % ? 

J7.  I own  sufficient  mining  stock,  par  $50,  paying  a semiannual  dividend  of 
2J%,  to  yield  me  $400  annually.  The  rate  of  brokerage  being  ^%,  and  the 
market  value  of  the  stock  being  $67^-,  how  many  U.  S.  3%  bonds  could  I get  if  I 
were  to  sell  the  stock  and  invest  the  proceeds  in  bonds  at  107|?  What  would 
be  my  surplus?  What  change  will  be  made  in  my  income? 


EXCHANGE 


739.  Exchange  is  the  method  or  system  by  which  debts  are  settled  between 
persons  in  different  places  without  the  actual  transmission  of  the  money.  It 
consists  in  the  giving  or  receiving  of  a sum  of  money  in  one  place  for  a bill 
ordering  the  payment  of  an  equivalent  sum  in  another. 

740.  Bank  drafts  are  the  principal  means  employed  by  merchants  in 
making  remittances  from  one  part  of  the  country  to  another.  Banks  located  in 
the  smaller  cities  and  towns  of  the  country  keep  money  deposited  in  the  great 
financial  centers,  such  as  New  York,  Chicago,  Boston,  Philadelphia,  St.  Louis, 
Baltimore,  New  Orleans  and  San  Francisco.  The  banks  draw  upon  their 
accounts  in  distant  cities  and  sell  their  drafts  to  their  customers,  making  a profit 
on  the  charge  for  “ exchange.”  They  also  buy  commercial  drafts,  drawn  bv  one 
merchant  upon  another  who  owes  him  money,  and  transmit  them  to  their  corre- 
spondents for  collection,  thus  keeping  their  accounts  replenished. 

741.  Drafts  upon  New  York  are  extensively  used  in  effecting  exchange 
between  the  other  financial  centers  of  the  country.  New  York  is  thus  the 
ultimate  center  of  exchange  for  the  whole  country. 

742.  The  principal  financial  centers  of  Europe  are  London.  Paris,  Amster- 
dam, Antwerp,  Hamburg,  Vienna,  Geneva,  Berlin,  Frankfort  and  Bremen. 
London  is  the  center  of  exchange  for  the  commerce  of  the  world. 

743.  A draft,  or  bill  of  exchange,  is  an  order  drawn  by  one  person  (the 
drciiver ) upon  another  (the  drawee)  living  in  a different  place,  directing  the  drawee 
to  pay  a sum  of  money  to  the  order  of  the  drawer  or  to  a third  person  (the  payee). 

744.  Bills  of  exchange  are  of  two  kinds — foreign  and  domestic. 

745.  A foreign  bill  of  exchange  is  one  payable  in  a country  other  than 
the  one  in  which  it  is  drawn. 

746.  A domestic,  or  inland  bill  of  exchange,  is  one  drawn  and  payable 
in  the  same  country. 

747.  When,  in  the  course  of  exchange  between  two  financial  centers,  the 
drafts  drawn  by  the  one  are  in  excess  of  those  drawn  by  the  other,  the  balances 
must  be  settled  by  shipment  of  actual  funds.  In  such  cases,  exchange  is  said  to 
be  against  the  one  place  and  in  favor  of  the  other ; and,  on  account  of  the  expense 
of  shipping  money,  drafts  drawn  in  the  one  city  (which  would  tend  to  increase 
such  expense)  are  at  a premium,  wdiile  drafts  drawn  in  the  other  city  (which 
would  tend  to  decrease  shipment  of  funds)  are  at  a discount. 


234 


EXCHANGE 


235 


748.  With  reference  to  time,  drafts  are  drawn  in  three  ways — at  sight,  a 
certain  number  of  days  after  sight,  and  a certain  number  of  days  after  date.  The 
first  are  called  sight  drafts ; the  other  two  are  called  time  drafts. 

749.  Time  drafts  are  discounted  in  the  same  manner  as  promissory  notes. 

750.  To  find  the  cost  of  a draft,  the  face  being  given. 

Example  1. — Find  the  cost  of  a draft  for  $1200,  exchange  \ % premium. 


$1200  face  of  draft,  Or, 
1.00J  cost  of  $1. 

1200.00 

3.00 

$1203.00  cost  of  draft. 


$1200  face. 

3 premium. 

$1203  total  cost. 


Example  2. — Find  the  cost  of  a draft  for  $800,  exchange  \%0  discount. 


100  Or,  $800  face. 

•00j  2 discount. 

$800X.99f  = $798  $798  net  cost. 


751.  Rule. — Multiply  face  by  par  {100%)  plus  the  rate  of  premium,  or  by  par 
less  the  discount.  Or,  Add  the  premium  to  the  face,  or  subtract  the  discount  from  the 
face. 

WRITTEN  PROBLEMS 


pr 


752.  Find  the  cost  of  the  following 

1.  $1000,  \ %0  discount. 

$2000,  \ °/0  premium. 

3.  $5000,  \ %0  discount. 

If.  $2385,  f %0  premium. 

5.  $3697,  discount. 

753.  To  find  the  face  of  a draft, 

Example. — What  is  the  face  of  a 
emium  ? 


drafts  : 

6 $3875.50,^%  premium. 

7.  $9217.84,  discount. 

8.  $7382.19,  yV  % premium. 

9.  $2468.36,  \%  discount. 

10.  $8696.78,  \ %0  premium. 

e cost  and  rate  being  given. 

draft  that  cost  $6015,  exchange  \% 


$1.00  face. 

.0025 

$1.0025  cost  of  $1. 


6000 

1.0025  ) 6015.0000 
6015  0 


face. 

entire  cost. 


000  Result  $6000. 


754.  Rule. — Divide  cost  by  par  {100%)  plus  the  rate  of  premium,  or  by  par  less 
the  discount. 


WRITTEN  PROBLEMS 

,755.  Find  the  face  of  the  following  drafts  (amount  given  being  the  cost,  and 
the  rate  expressed  in  dollars  and  cents  per  $1000)  : 


1.  $6234.18,  $1.25  premium. 

2.  $3697.25,  $2.50  discount. 

3.  $6800.00,  $1.00  premium. 

If..  $4675.39,  75  cents  discount. 
5.  $2500.00,  50  cents  premium. 


6.  $5675.00,  25  cents  discount. 

7.  $S322.12,  $2.00  premium. 

8.  $4725.80,  $1.75  discount. 

9.  $9786.45,  15  cents  premium. 

10.  $3375.95,  $1.25  discount. 


236 


EXCHANGE 


WRITTEN  PROBLEMS 

756.  Find  the  cost  of  the  following  drafts  : 

1.  A 30-day  draft  for  $5000,  \f0  premium. 

Explanation. — The  bank  discount  on  $1  of  the  draft  for  30  days  is  $.005  ; then  $1  of  the  face 
will  cost  $1 — $.005,  or  $.995  if  exchange  were  at  par ; but  at  premium  $1  of  the  face  will  cost 
$.995+$. 00125  or  $.99625,  and  $5000  will  cost  5000X$-99625  or  $.4981  25. 

2.  A 60-day  draft  for  $7500,  f % discount. 

3.  A 20-day  draft  for  $9873.25,  § % premium. 

A 12-day  draft  for  $6589.62,  \ % discount. 

5.  A 33-day  draft  for  $4437.18,  f % premium. 

Find  the  face  of  the  following  drafts  : 

6.  A 30-day  draft  that  cost  $7289,  $1.25  premium. 

7.  A 15-day  draft  that  cost  $6427.73,  discount. 

8.  A 60-day  draft  that  cost  $9745.56,  § % premium. 

9.  A 36-day  draft  that  cost  $4578.27,  50  cents  discount. 

10.  A 63-day  draft  that  cost  $5539.65,  f % premium. 

11.  I wish  my  agent  in  Chicago  to  buy  2000  bushels  of  wheat  at  68J  cents  a 
bushel.  If  his  commission  is  2%,  and  exchange  \ % premium,  what  will  be  the 
cost  of  the  sight  draft  I send  him  ? 

12.  A cotton  broker  in  New  York  sells  for  a dealer  in  New  Orleans  300  bales 
of  cotton  averaging  470  pounds  each  at  7§  cents  per  pound,  his  commission 
being  5%.  The  dealer  in  New  Orleans  draws  a 60-day  draft  for  the  amount  due 
him;  how  much  can  he  get  for  the  draft  if  exchange  is  $2.05  premium? 

13.  A of  New  York  owes  B of  San  Francisco  $15000.  If  exchange  is  f% 
premium,  how  much  must  he  pay  for  a 30-day  draft,  the  rate  of  interest  being 
6%  ? 

A t-  A man  in  St.  Louis  sends  to  a New  York  broker  500  shares  of  stock, 
with  instructions  to  sell  and  remit  proceeds  by  bank  draft.  If  the  broker  sells 
the  stock  at  87f,  and  exchange  is  \ % premium,  how  much  does  the  St.  Louis 
man  receive  ? 

15.  A commission  merchant  of  Boston  sold  a consignment  of  9S8200  pounds 
of  wheat  at  $1.20  per  bushel,  charging  2^%  commission,  and  $85  for  other 
expenses.  If  exchange  is  at  discount,  and  interest  6%  per  annum,  how 
large  a draft,  payable  20  days  after  sight,  can  he  buy  with  the  net  proceeds, 
allowing  4 days  for  the  acceptance  of  the  draft? 


FOREIGN  EXCHANGE 


757.  Foreign  exchange  is  the  settlement  of  debts  by  means  of  bills  drawn 
in  one  country  and  payable  in  another. 

758.  The  sum  of  a foreign  bill  of  exchange  is  expressed  in  the  money  of 
the  country  on  which  they  are  drawn. 

Note. — Foreign  bills  are  usually  drawn  at  sight  or  at  60  days  after  sight.  No  discount  is  reck- 
oned on  60-day  bills,  as  the  quotation  includes  the  allowance  for  time. 

759.  The  intrinsic  par  of  exchange  is  the  value  of  the  monetary  unit  of 
one  country  in  that  of  another,  based  on  the  comparative  weight  and  fineness  of 
the  coins  as  determined  by  government  assay. 

760.  The  commercial  par  of  exchange  is  the  market  value  of  the  coins  of 
one  country  when  sold  in  another. 

761.  The  rate  of  exchange  is  the  market  value  in  one  country  of  bills  of 
exchange  drawn  on  the  other.  While  this  value  depends  primarily  on  the 
balance  of  trade,  it  is  also  affected  by  the  ownership  in  one  country  of  the  invest- 
ment securities  of  another,  and  tbe  disbursement  of  interest  and  dividends.  The 
rate  can  never  rise  higher  than  the  actual  cost  of  exporting  or  importing  gold. 

762.  Commercial  quotations  of  foreign  exchange  are  given  by  means  of 
equivalents,  without  reference  to  the  par  value. 

763.  Sterling  exchange  is  quoted  by  giving  the  value  of  £l  in  dollars 
and  cents  ; as,  “ Sterling  exchange,  4.86J  @ 4.88  ” means  $4.86J  = £1  for  60-day 
bills  and  $4.88  = £1  for  sight  bills. 

764.  Exchange  on  France,  Belgium  and  Switzerland  is  quoted  by 
giving  the  value  of  $1  in  francs  and  centimes ; as,  “ Paris  exchange,  5.15  @ 
5.13^  ” means  $1  = 5.15  francs  for  60-day  bills  and  $1  — 5.13|  francs  for  sight 
bills. 

765.  Exchange  on  Holland  is  quoted  by  giving  the  value  of  1 guilder  in 
cents. 

766.  Exchange  on  Germany  is  quoted  by  giving  the  value  of  4 marks  in 
cents ; as,  94f  @ 94f  means  94f  cents  = 4 marks  for  60-day  bills  and  94f  cents 
= 4 marks  for  sight  bills. 

767.  It  is  provided  by  an  Act  of  Congress  that  the  values  of  the  standard 
coins  of  the  nations  of  the  world  shall  be  estimated  at  stated  intervals  by  the 
Director  of  the  Mint,  and  be  proclaimed  by  the  Secretary  of  the  Treasury.  The 
following  table  shows  the  values  of  the  foreign  moneys  of  account  as  published 
July  1,  1908. 


237 


238 


FOKEIGN  EXCHANGE 


Table  of  Foreign  Coins,  July  i,  1908 


VALUES  OF  FOREIGN  COINS 


COUNTRY 

Standard 

Monetary 

Unit 

Value  in 
terms  of 
U.  S.  gold 
dollar 

COINS 

Argentine  Republic  . . 

Gold  . 

Peso  . . . 

$0,965 

Gold  : argentine  ($4,824)  and  % argentine.  Sil- 
ver ; peso  and  divisions. 

Austria-Hungary  . . . 

Gold  . 

Crown  . . 

.203 

Gold.  10  and  20  crowns.  Silver:  land  5 crowns. 

Belgium  .... 

Gold  . 

I ranc 

.193 

Gold  : 10  and  20  francs.  Silver  : 5 francs. 

Bolivia 

Silver 

Boliviano 

.393 

Silver  : boliviano  and  divisions. 

Brazil 

Gold  . 

Milreis . . 

.546 

Gold:  5,  10  and  20  milreis.  Silver:  %,  1 and  2 
milreis. 

British  Possessions,  N. 

A.  (except  Newi’nd)  . 

Gold  . 

Dollar  . . 

1.000 

Central  Amer.  States— 

Costa  Rica 

Gold  . 

Colon  . . 

.465 

Gold  : 2,  5.  10  and  20  colons  ($9,307).  Silver  : 5, 
10,  25  and  50  centimos. 

British  Honduras . . 

Gold 

Dollar.  . . 

1.000 

Guatemala  ...  1 

Honduras ....  ( 

Nicaragua  ...  f 

Silver 

Peso  . . . 

.393 

Silver  : peso  and  divisions. 

Salvador  . . . j 

Chile 

China 

Gold  . 
Silver 

•{ 

Peso  . . . 

Tael  ||  . . 
Dollar  H . 

.365 

Gold:  escudo  ($1,825),  doubloon  ($.3650),  and  con- 
dor ($7,300).  Silver:  peso  and  divisions. 

Colombia 

Gold  . 

Dollar. . . 

1.000 

Gold : condor  ($9,647)  and  double-condor.  Sil- 
ver ; peso. 

Denmark 

Gold  . 

Crown  . . 

.268 

Gold  : 10  and  20  crowns. 

Ecuador  

Gold  . 

Sucre  . . 

.487 

Gold : 10  sucres  ($4.8665).  Silver : sucre  and 
divisions. 

Egypt 

Gold  . 

Pound 
100  piasters 

4.943 

Gold  ; pound  (100  piasters),  5,  10,  20  and  50  pias- 
ters. Silver  : 1,  2,  5,  10  and  20  piasters. 

Finland 

Gold  . 

Mark 

.193 

Gold  ; 20  marks  ($3,859),  10  marks  ($1.93). 

France  

Gold  . 

Franc  . . 

.193 

Gold:  5,  10,  20,  50  and  100  francs.  Silver:  5 
francs. 

German  Empire  . . . 

Gold  . 

Mark  . . . 

.238 

Gold  : 5,  10  and  20  marks. 

Great  Britain 

Gold 

Pound 

sterling 

4.866% 

Gold:  sovereign  (pound  sterling)  and  % sov- 
ereign. 

Greece 

Gold 

Drachma  . 

.193 

Gold  : 5,  10,  20,  50  and  100  drachmas.  Silver  : 5 
drachmas. 

Haiti  ...  

Gold  . 

Gourde  . . 

.965 

Gold:  1,  2,  5 and  10  gourdes.  Silver:  gourde  and 
divisions. 

India  (British) 

Gold  . 

Pound 

sterling* 

4.866% 

Gold:  sovereign  (pound  sterling).  Silver:  ru- 
pee and  divisions. 

Italy  

Gold  . 

Lira  . . . 

.193 

Gold  : 5,  10,  20,  5u  and  100  lire.  Silver  ; 5 lire. 

Japan  

Gold  . 

Yen  . . . 

.498 

Gold  : 5, 10  and  20  yen.  Silver:  10,  20  and  50  yen. 

Liberia 

Gold 

Dollar..  . 

1.000 

Mexico 

Gold  . 

Peso  f . . 

.498 

Gold:  5 and  10  pesos.  Silver:  dollar!  or  (peso) 
and  divisions. 

Netherlands 

Gold 

Florin  . . 

.402 

Gold  : 10  florins.  Silver  : 2%,  1 florin  and  divi- 
sions. 

Newfoundland 

Gold 

Dollar. . . 

1.014 

Gold  ; 2 dollars  ($2,028). 

Nor  wav 

Gold 

Crown  . . 

.268 

Gold  : 10  and  20-crowns. 

Panama 

Gold  . 

Balboa.  . 

1.000 

Gold : 1.  2%,  5,  10  and  20  balboas.  Silver  : peso 
and  divisions. 

Persia 

Silver 

Kran  . . . 

.072 

Gold  : %,  1 and  2 tomans  ($3,409).  Silver:  %,  %, 
1,  2 and  5 krans 

Peru 

Gold  . 

Libra.  . . 

4.866% 

Gold  : % and  1 libra.  Silver  : sol  and  divisions. 

Philippine  Islands  . . . 

Gold  . 

Peso  . . . 

.500 

Silver  peso  : 10.  20  and  50  centavos. 

Portugal 

Gold  . 

Milreis  . . 

1.080 

Gold  : 1,  2,  5 and  10  milreis. 

Russia 

Gold  . 

Ruble  . . 

.515 

Gold  : 5,  7%,  10  and  15  rubles.  Silver:  5,  10,  15,  1 
20,  25,  50  and  1U0  copeks. 

Spain  

Gold  . 

Peseta . . . 

.193 

Gold  : 25  pesetas.  Silver  : 5 pesetas. 

Straits  Settlements . . . 

Gold  . 

Pound 
sterling  § 

4.866% 

Gold:  sovereign  (pound  sterling).  Silver:  dol- 
lar and  divisions. 

Sweden 

Gold 

Crown  . . 

.268 

Gold  : 10  and  20  crowns. 

Switzerland 

Gold  . 

Franc  . 

.193 

Gold  : 5,  10,  20,  50  and  100  francs.  Silver : 5 : 
francs. 

Turkey  

Gold  . 

Piaster.  . 

.044 

Gold  : 25,  50, 100.  250  and  500  piasters. 

Uruguay  

Gold 

Peso  . . . 

1.034 

Gold  : peso.  Silver  : peso  and  divisions. 

Venezuela 

Gold  . 

Bolivar  . 

.193 

Gold : 5,  10,  20,  50  and  100  bolivars.  Silver : 5 
bolivars. 

Notes.— The  coins  of  silver-standard  countries  are  valued  by  their  pure  silver  contents,  at  the  average  market 
price  of  silver  for  the  three  months  preceding. 

*The  sovereign  is  the  standard  coin  of  India,  but  the  rupee  ($0.3244%)  is  the  current  coin,  valued  at  15  to  the 
sovereign. 

+Seventy-five  centigrams  fine  gold.  JValue  in  Mexico,  $0,498. 

gThe  current  coin  of  the  Straiis  Settlements  is  the  silver  dollar  issued  on  Government  account  and  has  been 
given  a tentative  value  of  $0.567758%. 

||The  value  of  the  tael  varies  in  the  different  provinces.  The  values  are  as  follows  : Amoy  $ 644,  Canton  .642, 
Cheefo'o  .616,  Chin  Kiang  .629,  Fuchau  .595,  Haikwan  (customs)  .655,  Hanko-v  .602,  Kiaochow  .624,  Nankin  .637, 
Niuchwang  .604,  Ningpo  .619,  Peking  .628,  Shanghai  .5S8,  Swatow  .595,  Takau  .648.  Tientsin  .624. 

HThe  dollar  has  the  following  values  in  the  respective  provinces  : Hongkdng  $.423,  British  .423.  Mexican  .427. 


FOREIGN  EXCHANGE 


239 


The  rates  in  the  foregoing  table  are  used  at  the  Custom  House  in  estimating 
the  value  of  foreign  merchandise  on  invoices  made  out  in  foreign  currencies. 

768.  To  find  the  cost  or  value  of  a foreign  bill. 

Example. — Find  the  value  of  a draft  for  £120  15s.  at  4.867. 

$4,867  4 ) 14601 

120f  36504 

584  040 
3 6504 
$587,690' 

769.  Rule. — Multiply  face  of  bill  by  quotation  of  exchange. 


WRITTEN  PROBLEMS 

770.  Find  the  value  in  United  States  money  of  the  following: 

6 £782  15s.  at  4.874. 


1.  £375  at  4.88. 

2.  2340  francs  at  5.15. 

3.  978  guilders  at  414. 
f.  1200  marks  at  964. 

5.  £562  8s.  6 cl.  at  4.864. 


7.  £524  7-s.  4 d.  at  4.86f. 

8.  7643.12  francs  a 5.134- 

9.  £1142  18s.  3d.  at  4.884. 

10.  3789  guilders  at  41|. 


771.  To  find  the  face  of  a foreign  bill. 

Example. — Find  the  face  of  a draft  on  London  costing  $587.69,  exchange 
4.867. 

£120  7499+ 


4.867  ) 587.690  ( 120 
486  7 
100  99 
97  34 

3 650 

20 

4.867  ) 7 3000  ( 15— 

4 867 
2 4330 
2 4335 


4.867  ) 587.690  0000 
486  7 
100  99 
97  34 


3 6500 
3 4069 


24310 

19468 

48420 

43803 

46170 


£ .7499 
20 

14.9980 
Result  £120  15s. 


Note. — The  exact  result  is  not  obtained  owing  to  the  ignoring  of  the  small  fraction  in  finding  cost. 

772.  Rule  . — Divide  vcdue  of  bill  by  quotation  of  exchange. 


WRITTEN  PROBLEMS 

773.  Find  the  face  of  hills  that  cost  as  follows  : 


1.  $738.72  at  4.S9  (London). 

2.  $975.18  at  5.144  (Paris). 

3.  $840.75  at  41f  (Amsterdam). 
f.  $1238.29  at  94§  (Berlin). 

5.  $3472.30  at  .517  (Antwerp). 


6.  $2575.84  at  4.85|  (London). 

7.  $395.97  at  4.88f  (London). 

8.  $1892.33  at  954  (Hamburg). 

9.  $5698.77  at  5.164  (Geneva). 

10.  $7535.44  at  96§  (Bremen). 


240 


TAXES 


WHITTEN  PROBLEMS 

774.  Find  the  value  of  the  following  hills  of  exchange  : 

1.  A draft  on  Vera  Cruz  for  3426  dollars,  at  .662. 

2.  A draft  on  Shanghai  for  6847  taels,  at  .901. 

3.  A draft  on  St.  Petersburg  for  13625  rubles,  at  .488. 

J.  A draft  on  Yokohama  for  24677  yen,  at  .658. 

5.  A draft  on  Havana  for  36355  pesos,  at  .926. 

6.  A Liverpool  merchant  buys  40000  bushels  of  American  wheat  at  69 
cents  a bushel ; how  much  does  it  cost  him  to  remit  by  draft  in  settlement, 
exchange  4.88J  ? 

7.  If  U.  S.  4s  are  quoted  in  New  York  at  112,  what  is  the  equivalent  London 
quotation,  exchange  4.88? 

Note. — American  securities  are  quoted  in  London  on  a fixed  basis  of  $5  = £1. 

8.  What  is  the  equivalent  New  York  quotation  of  stock  quoted  in  London 
at  87 f,  exchange  4.88J  ? 

9.  How  much  must  a London  merchant  pay  for  a draft  on  Bombay  of 
30000  rupees,  exchange  at  Is.  3f d.  (for  1 rupee)  ? 

10.  A cargo  of  wine  invoiced  at  23642.80  milreis  in  Lisbon  is  worth  more 
than  a cargo  of  coffee  invoiced  at  25000  milreis  in  Rio  Janeiro.  How  mueh 
more  in  U.  S.  money  ? 


TAXES 

775.  A tax  is  a sum  of  money  assessed  on  the  person  or  property  of  a citizen 
by  the  government  to  defray  public  expenses ; as,  a city,  borough  or  town  tax, 
county  tax,  state  tax,  etc. 

776.  The  public  revenues  of  the  various  state  and  local  governments  in  the 
United  States  are  raised  by  direct  taxation  upon  the  property,  and  in  some  of  the 
states  upon  the  polls  and  personal  incomes. 

777.  The  public  revenues  of  the  National  government  are  raised  by  indirect 
taxation  in  the  form  of  duties  on  imported  goods  and  the  internal  revenue  taxes 
on  liquors  and  tobacco. 

778.  A capitation  or  poll  tax  is  a specified  sum  levied  on  the  person  of 
every  adult  male  citizen. 

779.  Property  taxes  are  divided  into  two  classes — (a)  taxes  on  real  property 
or  real  estate , i.  e.,  lands,  houses,  etc.;  and  (b)  taxes  on  personal  property  or 
personal  estate,  i.  e.,  horses,  cattle,  vehicles,  furniture,  money,  stocks,  bonds, 
mortgages,  merchandise,  etc. 


TAXES 


241 


780.  To  find  amount  of  tax. 

Example  — What  is  the  amount  of  tax  on  a property  assessed  at  $2250, 
rate  13  mills? 

$2250 

.013 

$29,250 

781.  Rule  . — Multiply  valuation  by  rate , expressed  decimally. 


ORAL  EXERCISE 

782.  1.  What  is  the  tax  on  $2500,  at  the  rate  of  9 mills  on  the  dollar? 

2.  On  $17000,  at  4 mills? 

3.  On  $4000,  at  6J  mills? 

4.  On  $24000,  at  8 ^ mills  ? 

5.  What  is  the  tax  on  $3000,  at  the  rate  of  $1.75  on  $100  ? 

6.  On  $4000,  at  $1.50  ? 

7.  On  $12000,  at  $1.25? 

8.  What  is  the  tax  on  $20000,  at  the  rate  of  $18.50  on  $1000  ? 

9.  On  $11000,  at  $17? 

10.  On  $6875,  at  $16? 

783.  Tax  Duplicate 


Valuation. 

1. 

$5000 

2. 

$3000 

3. 

$2500 

4- 

$12500 

5. 

$8000 

6. 

$6500 

7. 

$800 

8. 

$7250 

9. 

$4300 

10. 

$1350 

11. 

$950 

12. 

$4000 

13. 

$2000 

n. 

$1700 

15. 

$3900 

County  Tax 

Road  Tax 

School  Tax 

Poor  Tax 

Total. 

15  mills 

8 mills. 

10  mills. 

2 mills. 

242 


TAXES 


784.  To  find  rate  of  tax  required  to  raise  given  amount  of  tax. 

Example. — Required  tax  rate  on  assessed  valuation  of  $1700000  to  raise  a 
school  tax  of  $18000,  10%  of  levy  being  uncollectable,  and  cost  of  collection 
being  2%. 

$1.00  of  tax  levied.  .882  )18000.0000(  20408.163  total  amount 

.10  of  tax  un  collectable. 

.90  of  tax  collectable. 

.018  amount  paid  collector  on  each 
$1  of  tax  levied. 

.882  net  amount  received  on  each 
$1  of  tax  levied. 

1700000  ) 20408.16  ( .012 
17000  00 

3408  16 

3400  00  Result  12  mills. 

The  result  is  nearly  12  mills,  which  would  be  the  rate  to  levy  to  cover  the 
amount  needed. 

785.  Rule. — Divide  amount  necessary  to  levy  by  assessed  valuation. 

WRITTEN  PROBLEMS 

786.  1.  A city  whose  assessed  valuation  is  $2500000  must  raise  830000  for 
city  purposes  ; $12500  for  highways;  $20000  for  schools;  $5000  for  the  support 
of  the  poor;  these  sums  include  collector’s  fee.  What  rate  must  be  levied  for 
each  purpose? 

2.  What  is  the  tax  on  $14500  of  real  estate,  and  $23275  of  personal  prop- 
erty, at  $2.07  per  $100,  less  a discount  of  2%  for  prompt  payment? 

3.  If  A’s  tax  is  $541.64,  at  the  rate  of  $19.75  on  $1000,  what  is  the  assessed 
valuation  of  his  property  ? 

It.  What  is  the  rate  of  taxation  per  $100,  if  property  assessed  at  $3850  pays 
a tax  of  $73.42  ? 

5.  The  assessed  valuation  of  all  the  taxable  property  in  a certain  town  is 
$1987690.  The  number  of  polls  is  418,  at  50  cents  each.  The  estimated  expenses 
of  the  town  are  $33210.  What  rate  of  taxation  will  raise  this  amount  (assuming 
that  all  the  taxes  can  be  collected)  ? 

6.  What  must  be  the  rate  of  taxation  to  yield  $5289.70,  after  paying  col- 
lector’s commission  of  2%,  if  the  assessed  value  of  the  property  is  $321500? 

Note. — The  tax  colleotor  receives  his  commission  upon  the  gross  amount  of  tax  collected. 

7.  Mr.  Brown  was  assessed  as  follows  : Real  estate,  $30000  ; personal  prop- 
erty, $3500 ; money  at  interest,  $25000;  income  from  occupation,  $3000;  and 
two  gold  watches.  He  obtains  an  abatement  of  one-third  on  real  estate,  one- 
fourth  on  personal  property,  $3000  on  money  at  interest,  two-fifths  for  occupation, 
and  one  gold  watch.  The  tax  rate  was  2J  mills,  and  $1.50  for  each  watch.  What 
was  Brown’s  tax  after  the  abatement,  and  how  much  was  it  lessened  ? 


1764  to  be  levied. 

3600 

3528 

7200 

7056 

1440 

882 

5580 

5292 

2880 

2646 


DUTIES 


787.  Duties  or  customs  are  taxes  levied  by  the  Government  upon  imported 
goods  for  revenue  for  the  support  of  the  general  government  and  for  the  protec- 
tion of  home  industries. 

788-  Duties  are  of  two  classes — ad  valorem  duties  and  specific  duties. 

789.  An  ad  valorem  duty  is  a tax  assessed  at  a certain  per  cent,  on  the 
cost  of  the  goods  in  the  country  from  which  they  are  imported. 

Note. — Ad  valorem  duties  are  computed  on  the  invoice  cost  of  goods  “ packed  and  ready  for  ship- 
ment,” exclusive  of  subsequent  expenses,  such  as  freight,  insurance,  etc.  In  custom  house  calculations, 
duties  are  not  reckoned  on  fractions  of  a dollar  ; a fraction  of  a dollar  if  less  than  one-half  is  rejected, 
if  over  one-half  is  counted  as  another  dollar. 

790.  A specific  duty  is  a tax  assessed  at  a certain  sum  per  pound,  ton, 
gallon,  foot,  yard,  or  other  weight  or  measure,  without  regard  to  value. 

791.  Tare  is  an  allowance  made,  in  estimating  specific  duties,  by  way  of 
deduction  from  the  gross  weight  of  goods  on  account  of  the  weight  of  the  box, 
cask,  etc.,  in  which  they  are  contained. 

792.  Leakage  is  an  allowance  made  for  waste  of  liquids  in  barrels  or  casks. 

793  Breakage  is  an  allowance  for  waste  of  liquids  in  bottles. 

794.  To  find  ad  valorem  duty. 

/3S3 

72  & 3 0 

2 f / O //  OS,  f 

3 3 tr  r o 

/3  S 3.3  07  f 

Note. — Take  duty  on  dollars  only,  dropping  cents  if  less  than  50,  adding  $1  if  50  cents  or  more. 

795.  Rule. — Find  value  of  goods  in  United  States  money  and  multiply  by  rate 
of  duty. 

Note. — Specific  duty  is  simply  a tax  on  quantity  and  is  found  accordingly.  Ad  valorem  duty 
is  always  expressed  at  a certain  rate  per  cent. 

WRITTEN  PROBLEMS 

796.  1.  What  is  the  duty  on  an  invoice  amounting  to  12673.10  francs  at 
40%  ad  valorem  ? 

2.  Find  the  duty  at  7 cents  per  pound  on  an  invoice  of  goods  weighing 
32478  lbs.,  tare  2%. 


Example. — What  is  the 
duty  on  an  invoice  of  goods 
invoiced  at  £72  12s.  at  30% 
ad  valorem  ? 


243 


244 


DUTIES 


3.  What  is  the  duty  on  an  importation  of  goods  invoiced  at  12643  marks  at 
40%  ? 

Ip.  Find  the  duty  on  goods  invoiced  at  7846  francs  at  60%. 

5.  At  50%,  what  is  the  duty  on  an  invoice  amounting  to  6955  lire? 

6.  Find  the  duty  on  goods  invoiced  at  23632  florins  at  25%. 

7.  Find  the  duty  on  an  invoice  of  £593  17s.  10d.  at  45%. 

8.  At  35  cents  per  gallon,  what  is  the  duty  on  100  cases  of  olive  oil,  each 
case  containing  2 dozen  quarts? 

9.  Find  the  duty  on  40  blocks  of  marble,  each  1J  X 2 X 4 ft.  at  50  cents  per 
cubic  foot. 

10.  At  10%,  what  is  the  duty  on  an  invoice  of  12684  lbs.  of  leather,  im- 
ported from  Liverpool  at  Is.  2 ^d.  per  pound? 

11.  At  35%,  what  is  the  duty  on  an  invoice  of  chinaware  from  Paris,  valued 
at  14274  francs? 

12.  Find  the  duty  on  300  dozen  penknives,  valued  at  3s.  per  dozen,  at  25 
cents  per  dozen  and  25%  ad  valorem;  and  400  dozen  penknives,  valued  at 
4s.  6d.  per  dozen,  at  40  cents  per  dozen  and  25%  ad  valorem. 

13.  What  is  the  duty  on  26548  lbs.  of  tobacco  at  40  cents  per  pound  ? 

lip  Find  the  duty  on  14400  spools  of  cotton  thread  at  5J  cents  per  dozen 
spools  ? 

15.  At  30  cents  per  square  yard,  what  is  the  duty  on  5694  yards  of  cloth  27 
in.  wide? 

16.  At  25%,  find  the  duty  on  an  importation  of  goods  from  Buenos  Ayres, 
invoiced  at  7642  pesos. 

17.  At  30%,  find  the  duty  on  an  importation  of  goods  from  Havana, 
invoiced  at  12638  pesos. 

18.  At  40%,  find  the  duty  on  an  importation  of  goods  from  Montevideo, 
invoiced  at  9847.60  pesos. 

19.  At  20%,  find  the  duty  on  an  importation  of  goods  from  Valparaiso, 
invoiced  at  32624.80  pesos. 

20.  At  30%,  find  the  duty  on  an  importation  of  goods  from  Bogota,  invoiced 
at  2978.25  pesos. 

21.  What  is  the  duty  on  an  importation  of  goods  from  Paris,  invoiced  at 
38734  milreis  at  35%  ? 

22.  What  is. the  duty  on  an  importation  of  goods  from  Oporto,  invoiced  at 
19875  milreis  at  45%  ? 

23.  What  is  the  duty  on  an  importation  of  goods  from  Shanghai,  invoiced 
at  24653  taels  at  30  % ? 

2 Ip.  What  is  the  duty  on  an  importation  of  goods  from  Yokohama,  invoiced 
at  64882  yen  at  25%  ? 

25.  What  is  the  duty  on  an  importation  of  goods  from  Constantinople 
invoiced  at  527643  piasters  at  20%  ? 


RATIO  AND  PROPORTION 


245 


36.  Make  the  extensions  and  calculate  the  duty  on  the  following  invoice : 


876J  yds.  Broadcloth 
278f  “ 


Discount  2 % 

Boxes  and  packing 
Cartage 


Duty,  50  %. 


15s.  8 d. 
IS s.  6d. 


£ 

s. 

17 

6 

d. 


RATIO  AND  PROPORTION 

RATIO 

797.  Ratio  is  a measure  of  relation  between  quantities  of  the  same  kind 
There  are  two  kinds  of  ratio — arithmetical  and  geometrical. 

798.  Arithmetical  ratio  expresses  the  difference  between  two  quantities. 

799.  Geometrical  ratio  is  the  division  of  one  term  by  another.  The  usual 
way  of  expressing  a geometrical  ratio  is  by  placing  two  points,  one  above  the 
other,  between  the  quantities  compared.  Thus  2 : 3 signifies  the  ratio  of  2 to  3, 
and  is  read  2 is  to  3.  The  quantities  compared  are  called  the  tervis  of  the  ratio. 
The  first  is  called  the  antecedent,  the  second  the  consequent. 

800.  The  value  of  a ratio  is  found  by  dividing  the  antecedent  by  the  conse- 
quent. Since  the  antecedent  of  the  ratio  is  always  the  dividend  and  the  conse- 
quent the  divisor,  the  ratio  may  be  written  in  the  form  of  a fraction,  the  antecedent 
being  the  numerator  and  the  consequent  the  denominator.  It  follows  that  the 
terms  of  a ratio  may  be  multiplied  or  divided  in  the  same  manner  as  the  terms 
of  a fraction,  without  changing  the  value  of  the  ratio. 

801.  A simple  ratio  consists  of  one  antecedent  and  one  consequent ; as, 
10  : 12. 

802.  A compound  ratio  consists  of  two  or  more  simple  ratios  ; as, 
9 : 12  | 

8 : 14  i ‘ 

803.  The  value  of  a compound  ratio  is  found  by  dividing  the  product 
of  the  antecedents  by  the  product  of  the  consequents  ; thus,  in  the  above  illus- 
tration, 

3 2 

9 X 'f>  3 

7 

P 7 


246 


RATIO  AND  PROPORTION 


804.  To  find  the  value  of  a simple  ratio. 

Example. — What  is  the  ratio  of  $1000  to  $300? 

$1000  : $300  - = ty-  or  3*. 

805.  Rule. — Divide  the  antecedent  by  the  consequent. 


ORAL  EXERCISE 


806.  1.  What  is  the  ratio  of  9 : 6 ? 

9 : 6 = f or  11. 

Find  the  values  of  each  of  the  following  ratios: 
2.  5 : 8.  5.  15  : 5.  8.  25  : 6. 

11.  4f  : 2|. 

3.  7:6. 

6.  3J  : 2R  9.  151  : 4J. 

IS.  8i  : 171 

4.  12  : 26. 

7.  17  : 4.  10.  6i  : 9|. 

13.  5f  : 14f. 

14.  $8  : $10. 

horses  horses 

mo.  da.  mo.  da. 

men  men 

17.  54  : 18. 

19.  4 6 : 8 12 

15.  27  : 9. 

bu.  bu. 

mi.  mi. 

yd.  yd. 

18.  3J  : 20. 

20.  25  : 75. 

16.  16f  : 33 J. 

MENTAL  PROBLEMS 

807.  1 . A pole  60  feet  long  is  sunk  15  feet  in  the  earth  ; what  part  of  it  is  in 
the  air  ? 

Solution. — The  part  in  the  earth,  15  feet,  is  1 of  60  feet,  the  length  of  the  pole  ; hence  \ of  the 
pole  is  in  the  air. 

2.  Two  men  engage  in  a business  venture,  the  gains  or  losses  to  be  shared 
in  proportion  to  money  furnished  ; A puts  in  $60  and  B $90  ; how  should  they 
share  the  gain  ? 

3.  In  a mixture  of  clover  and  timothy  30  pounds  are  clover  and  10  pounds 
are  timothy;  what  is  the  proportion  of  each  in  the  mixture? 

J.  In  a blend  of  coffee  39  pounds  are  Java  and  13  are  Mocha ; what  is  the 
proportion  of  each  in  the  blend  ? 

5.  A farmer  made  an  insecticide  of  99  parts  water  and  1 part  London 
purple  ; what  is  the  ratio  of  the  London  purple  to  the  water?  Of  the  water  to 
the  mixture  ? 

6.  If  Java  and  Mocha  are  blended  in  the  proportion  of  4 to  1,  how  many 
pounds  of  each  in  a mixture  of  60  pounds? 

7.  In  a.  blend  of  tea,  the  black  is  to  the  green  as  5 to  3 ; how  many  pounds 
of  each  in  a mixture  of  24  pounds  ? 

8.  In  a certain  school,  the  boys  are  to  the  girls  as  8 to  5 ; how  many  of  each, 
the  total  number  on  roll  being  52? 

9.  Doe’s  money  is  to  Roe’s  as  7 to  5 ; how  much  has  each  if  both  have  $96  ? 

10.  In  a mucilage  mixture  3 pounds  of  water  are  used  and  4 pound  of  gum- 

arabic;  how  much  of  each  must  be  taken  to  make  70  pounds? 


RATIO  AND  PROPORTION 


247 


PROPORTION 

808-  A proportion  is  the  expression  of  equality  between  two  ratios. 
Proportion  is  indicated  by  placing  four  points  between  the  ratios — thus, 
6 : 12  : : 24  : 48. 

809.  The  first  and  last  terms  of  a proportion  are  called  the  extremes  ; 
the  second  and  third,  the  means. 

810.  In  any  proportion,  the  product  of  the  extremes  equals  the  product  of  the 
means.  It  follows  from  this  that  either  extreme  is  found  by  multiplying  the 
means  and  dividing  this  product  by  the  given  extreme;  or,  either  mean  is  found 
by  multiplying  the  extremes,  and  dividing  by  the  given  mean. 

811.  A simple  proportion  expresses  the  equality  of  two  simple  ratios. 

812.  To  find  the  fourth  term  of  a proportion. 

Example  1 — 25  : 42  : : 70  : ? 

14 

42  X ,70  588 

=—  = 11 

Ip  o 

5 

Example  2—  § : : : ££  : ? 

2 0 
8X11X21  1 

2 

,4X22 

2 

Note — Since  tlie  first  term  is  the  divisor,  we  simply  invert  it  and  multiply. 

813.  Rule. — Divide  the  product  of  the  means  by  the  first  term. 

Note. — The  operation  may  frequently  be  abridged  by  cancelation. 


814. 


WRITTEN  EXERCISE 


1. 

17 

: 32  : : 

51  : ? 

7. 

7 : 4i  : 

: 94  : ? 

2. 

1 5 
1 6 

. 7 . . 

• 64  • ’ 

f :? 

8. 

15  : 27  : 

: 8 : ? 

3. 

9 : 

24  : : 

7 : ? 

9. 

144  : 5 : 

: 94  : ? 

4- 

33 

: 46  : : 

3 : ? 

10. 

3 : 7i  : 

: 21  : ? 

5. 

5 : 

7 : : 9 

: ? 

11. 

4 : 26  : : 

: 9 : ? 

6. 

74 

: 134: 

:3J:? 

12. 

7f  : 15 

• 73.9 

815.  To  determine  the  statement  of  a simple  proportion. 

Example  1. — If  15  men  can  do  a piece  of  work  in  22  days,  in  how  many 
days  can  24  men  do  the  same  work  ? 


If  15  men  can  do  the  work  in  22  days,  24  men  can  do  the  work  in  less  time  ; in  this  case  the 
smaller  term  of  the  complete  ratio  is  placed  as  the  second  term.  The  proportion  is  24  : 15  : : 22  : ? 

5 11 

pX22  55 

~ = — or  13f  days. 

2£  4 J 


4 


248 


RATIO  AND  PROPORTION 


Example  2.  If  19  men  can  earn  $410  in  a given  time,  how  much  could 
28  men  earn  in  the  same  time? 


If  19  men  can  earn  $410,  28  men  can  earn  more  ; in  this  case  the  larger  term  of  the  complete 
ratio  is  placed  as  the  second  term.  The  proportion  is  19  : 28  : : 410  : ? 

28X410  11480 


19 


19 


or  $604.21 


816.  R ule. — Place  the  term  which  is  of  the  same  kind  as  the  required  result 
as  the  third  term.  Then  from  the  conditions  of  the  problem,  determine  whether  the 
fourth  term,  or  result,  should  he  more  or  less  than  the  given  third  term.  If  the  fourth 
term  should  be  less  than  the  third , place  the  smaller  of  the  remaining  terms  second ; if 
more,  place  the  larger  of  the  remaining  terms  second. 


WRITTEN  PROBLEMS 

817.  1.  If  29  tons  of  coal  cost  $130.50,  what  would  35  tons  cost  at  the  same 
rate  ? 

2.  If  32  horses  can  do  a given  amount  of  work  in  46  days,  in  what  time 
could  54  horses  do  the  same  amount  of  work  ? 

3.  If  $326  produce  $1980  interest  in  a given  time,  how  much  interest 
would  $1920  produce  in  the  same  time  at  the  same  rate? 

f.  If  $49.75  is  the  interest  of  a given  principal  at  5%,  what  is  the  interest 
on  the  same  principal  for  the  same  time  at  1\%  ? 

5.  If  $462  produce  $29.75  interest  in  a given  time,  what  principal  for  the 
same  time  and  rate  will  produce  $92.64  interest? 

6.  If  15f-  pounds  of  butter  cost  $5.04.  what  will  23§  pounds  cost  at  the 
same  rate  ? 

7.  If  the  liabilities  of  a bankrupt  amount  to  $11726,  and  the  assets  to  89427, 
how  much  should  A receive  to  whom  he  owes  $1247  ? 

Note. — The  entire  liability  is  to  the  liability  of  each  creditor  as  the  entire  assets  are  to  that 
creditor’s  share. 

8.  The  collectable  accounts  of  an  insolvent  merchant  are:  notes,  84675 ; 
personal  accounts,  $11625 ; he  has  $2362  worth  of  stock  on  hand.  His  liabilities 
are  : notes,  $13654  ; and  other  debts  amounting  to  $15065.  How  much  can  he 
pay  A to  whom  he  owes  $1290;  B,  to  whom  he  owes  $2265,  and  C,  to  whom  he 
owes  $4475  ? 

9.  The  total  assessment  of  a borough  is  $3457500  ; the  tax  to  be  raised  for 
1909  is  $14750.  How  much  of  this  should  a property  holder  pay  whose  property 
is  assessed  at  $18500? 

Note. — The  total  assessment  is  to  each  individual  assessment  as  the  tax  to  be  raised  is  to  the 
amount  to  be  paid  by  that  individual. 

10.  The  total  tax  to  be  raised  by  a certain  town  for  the  year  1908  is  $17630. 
The  assessed  valuation  of  the  town  is  $4724000.  What  tax  should  A pay  whose 
property  is  assessed  at  $7500 ; B whose  property  is  assessed  at  $12300 ; and  C 
whose  property  is  assessed  to  the  amount  of  $17500? 


RATIO  AND  PROPORTION 


249 


11.  A and  B form  a partnership.  A invests  $12500  and  B invests  $16750. 
They  gain  during  the  year  $7250.  Find  each  partner’s  share  of  the  gain. 

Note. — The  total  investment  is  to  each  partner’s  investment  as  the  whole  gain  is  to  each 
partner’s  gain. 

12.  Three  boys  buy  a bicycle,  each  paying  as  follows  : the  first  $36,  the  sec- 
ond $45,  the  third  $49.  How  long  should  each  use  it  every  day  of  ten  hours? 
How  long  should  each  keep  it  in  a period  of  3 weeks? 

13.  If  § of  a yard  of  silk  cost  $1.85,  what  should  If  yards  cost  at  the  same 
rate? 

14-.  If  a merchant  gains  16§  % by  selling  goods  for  $14,  what  would  be  his 
gain  per  cent,  by  selling  them  for  $17  ? 

Suggestion. — The  selling  price  contains  cost  and  gain,  hence,  before  a proportion  can  he  formed, 
the  proper  conditions  must  be  obtained. 

Remark. — The  student  should  carefully  observe  the  conditions  of  problems  to  see  that  they  are 
proportional.  Some  of  these  problems  are  formed  with  the  object  of  testing  the  student’s  percep- 
tion in  this  respect,  and  in  others,  the  operation  of  proportion  is  only  incidental  and  something  remains 
to  be  done  to  obtain  the  required  result.  This  is  very  frequently  the  case  in  the  actual  arithmetic 
of  business. 

15.  If  by  selling  goods  for  f of  the  marked  price  I gain  f of  the  cost,  what 
part  of  the  cost  would  I gain  by  selling  them  for  f of  the  marked  price? 

16.  If  25  men  can  do  a piece  of  work  in  34  days,  in  what  time  could  42  boys 
do  the  same  work,  if  7 boys  can  do  as  much  as  5 men  ? 

17.  A works  15  hours  at  a certain  job,  B 22  hours,  C 28  hours.  If  the 
amount  received  for  the  work  is  $42.50,  what  should  each  receive? 

18.  A traveler  purchased  a railroad  ticket  for  $24.  To  get  to  his  destination 
he  travels  over  125  miles  of  one  company’s  road,  216  miles  of  another  and  475 
miles  of  a third.  Find  the  sum  due  each  road. 

19.  If  42  men  or  54  boy's  can  do  a piece  of  work  in  24  days,  how  long  would 
it  take  18  men  and  26  boys  to  do  the  same  work  ? 

20.  If  44f  yards  of  cloth  cost  $145.03,  what  should  37f-  yards  cost? 

21.  If  the  freight  on  a car  load  of  coal  from  one  point  to  another  is  $8.75, 
and  it  is  carried  36 J miles  on  one  company’s  road  and  46-J  miles  on  another, 
what  share  of  the  freight  should  each  receive? 

22.  If  by  selling  goods  for  $97  there  is  a loss  of  15%,  what  would  be  the 
per  cent,  of  loss  or  gain  by  selling  them  for  $102  ? 

23.  If  a grocer’s  pound  weight  is  f of  a pound  light,  what  would  his  sales 
amount  to  in  a week,  if  during  this  time  he  has  defrauded  his  customers  of  $52? 

21^.  If  I buy  coal  for  $4.25  a long  ton,  and  wish  to  sell  by  the  short  ton  so  as 
to  gain  25%,  what  must  I ask  per  ton? 


250 


KATIO  AND  PROPORTION 


COMPOUND  PROPORTION 

818.  To  find  the  value  of  a compound  ratio. 


Example. — What  is  the  value  of 


9 : 6 


6 : 18 
?X0  1 

2 


2 

819.  Rule. — Divide  the  product  of  the  antecedents  by  the  product  of  the 
consequents. 

ORAL  EXERCISE 

820.  Find  the  value  of  the  following  compound  ratios : 


1. 


4 : 6 

5 : 9 


l. 


2. 


4* 


3. 


8 : 12 
15  : 9 
74:14 
15  : 4 
8 : 12 


r 

\ 

r 


5. 

6f  : 

CO 

124  : 

CO 

5 : 

8 

6. 

7 : 

6 

men 

men 

6 : 

8 

7. 

da. 

da. 

5 : 

6 

$4: 

SI  2 

8. 

hr. 

hr. 

8 : 

10 

9. 


10. 


11. 


12:7  1 
acres  acres  / 

34  : 6 J 


12. 


rods  rods 

12  : 15 


men  men 

8 : 6 


821.  A compound  proportion  is  the  expression  of  equality  between  ratios, 
one  of  which  is  compound.  Thus: 

4:5  v 

8 : 12  l 

35  : 15  j 


25  : 20.V 


Note. — The  fourth  term  in  a compound  proportion  is  found  by  multiplying  the  means,  and 
dividing  this  product  by  the  product  of  the  extremes. 


5X42X  15x25  . 

8X#  — 


822.  To  find  the  fourth  term  in  a compound  proportion. 


Example.- 


3f  : 44 


8 

134 


9 

13# 


32 . ? 

°9  ’ • 


4xSx  2x9x5x68x29 


15x7x27x2x9X  5X  9 


012074 
“'25  5 15 


RATIO  AND  PROPORTION 


251 


823.  Rule. — Multiply  together  the  means,  and  divide  by  the  product  of  the 
given  extremes. 


WRITTEN  EXERCISE 

824.  Find  the  fourth  term  in  each  of  the  following  : 


1. 


4 : 9 

7 : 6 

8 : 5 


1 


20  : 21  J 


: : 9 : ? 


9 : 14  ) 

7:6  > : : 24  : ? 

15: 36  i 


3. 


4- 


74  : 35  1 

3i  : 12  - : : 6 :? 

5f  : 22  ) 

14:26  1 

8:28  l • • 71 

7J  : 45  { ' ' ‘2 

3|  : 15  j 


14  : 26 
25  : 65 

5.  33  : 72  ■ 

25  : 40  I 
125  : 140  J 

44:18  ) 

6-  5i  : 16  V 
6f  : 42  J 


: : 16  :? 


: : 15  : ? 


825.  To  determine  the  statement  of  a compound  proportion. 

Example. — If  14  men,  working  9 hours  a day,  can  build  a wall  27  feet  long, 
18  inches  thick  and  28  feet  high  in  18  days,  in  how  many  days  of  10  hours  each 
could  22  men  build  a wall  38  feet  long,  2 feet  thick  and  12  feet  high  ? 


Statement  : 

22  : 14  1 
10  : 9 

27  : 38 
18  : 24 

28  : 12 


f 


1 


14 

22 


: : 18  :? 


brs. 

9 

10 


long 

27  ft. 
38  ft. 


thick  high  da. 

18  in.  28  ft.  18 

2 ft.  12  ft.  ? 


2 19  3 

474X9X^X24  Xf2x,18 

22x49X2, 7x4$X2$ 

11  5 9 J[ 


826.  Rule. — Write  as  the  third  term  the  quantity  which  is  of  the  same  kind  as 
the  result.  Then,  as  in  simple  proportion,  determine  whether  the  answer  depending 
upon  each  ratio  should  be  more  or  less  than  the  third  term  ; if  more,  place  the  larger 
term  of  ratio  nearest  tlte  third  term  ; if  less,  place  the  smaller  term  of  the  ratio  nearest 
the  third  term.  Proceed  in  this  way,  reasoning  with  each  ratio  entirely  independent 
of  the  others,  until  all  have  been  disposed  of. 


WRITTEN  PROBLEMS 

827.  1.  If  the  interest  of  $367  at  5 % for  4 years,  7 months,  18  days,  is  $85.02, 
what  is  the  interest  of  $19.72  at  4|%  for  2 years,  5 months  and  18  days  ? 

2.  If  the  freight  on  32  barrels  of  sugar  for  a distance  of  54  miles  is  $4.25, 
what  should  be  the  freight  on  63  barrels  for  28  miles  ? 

3.  If  a 3-pound  loaf  of  bread  cost  10  cents  when  flour  is  worth  $4.75  a 
barrel,  what  should  be  the  price  of  a 4-pound  loaf  when  flour  is  worth  $5.85 
a barrel  ? 


252 


RATIO  AND  PROPORTION 


If  15  apples  can  be  bought  for  25  cents  when  apples  are  worth  §4.25  a 
barrel,  what  should  be  the  cost  of  45  apples  when  a barrel  costs  §4? 

5.  If  the  tax  on  a property  assessed  at  $2700  is  $128.25  at  a tax  rate  of  4f  %, 
what  should  be  the  tax  on  a property  assessed  at  $3800  if  the  rate  is  2J%  ? 

6.  If  I pay  $2.25  a yard  for  cloth  f of  a yard  wide,  what  should  I pay  for 
22  yards  of  a similar  quality  J of  a yard  wide  ? 

7.  If  26  men,  working  10  hours  a day,  can  dig  a ditch  142  feet  long,  4 feet 
wide  and  12  feet  deep  in  96  days,  in  how  many  days  of  8 hours  each  could  92 
men  dig  a ditch  108  feet  long,  3 feet  wide  and  16  feet  deep,  if  the  digging  in  the 
second  case  is  estimated  to  be  20%  harder? 

8.  If  4 horses,  working  9 hours  a day,  can  plow  a field  372  feet  long,  348 
feet  wide,  in  2J  days,  the  walking  gait  of  the  horses  being  4 miles  an  hour,  in 
bow  many  days,  working  10  hours  a day,  could  6 horses,  whose  walking  gait  is 
3 miles  an  hour,  plow  a field  containing  14  acres,  142  perches? 

9.  If  $765  produce  $144.59  interest  in  2 years,  8 months  and  12  days  at 

7 %,  what  principal  would  produce  $146.75  in  3 years,  2 months  and  18  days  at 

5%  ? 

10.  If  $1365,  in  3 years,  9 months  and  14  days,  at  6%,  produce  $310.31 
interest,  at  what  rate  would  $1065  produce  $78.41  in  1 year,  7 months  and  19 
days? 

11.  If  $1426,  at  5%,  will  produce  $163.99  interest  in  2 years,  3 months  and 
18  days,  in  what  time  will  $1820  produce  $726.80  interest  at  44%  ? 

12.  If  a pile  of  wood  18  feet  long,  4 feet  wide  and  9 feet  high  cost  $22.50, 
what  should  be  paid  for  a pile  21  feet  long,  3 feet  wide  and  11  feet  high  ? 

13.  If  I pay  $42.25  for  a pile  of  wood  16  feet  long,  5 feet  wide  and  7 feet 
high,  how  wide  should  a pile  be  to  be  worth  $50.50  if  it  is  14  feet  in  length  and 

8 feet  high  ? 

14-.  If  there  are  20  perches  of  masonry  in  a wall  33  feet  long,  5 feet  high 
and  3 feet  thick,  bow  many  perches  are  there  in  a wall  32  feet  long,  14  feet  high 
and  2 feet  thick? 

15.  If  a wall  122  feet  long,  12  feet  high,  2|  feet  thick  contains  148  perches 
of  masonry,  how  high  should  a wall  136  feet  long  and  3 feet  thick  be  to  contain 
136  perches? 

16.  How  many  Belgian  blocks,  3J  inches  wide,  7 inches  long,  will  be 
required  to  cover  a space  36  feet  wide,  88  feet  long,  if  5120  blocks,  9 inches  long, 
4J  inches  wide,  are  required  to  cover  a space  32  feet  wide  and  45  feet  long? 

17.  How  many  hours  a day  must  64  men  work  to  dig  a canal  2 miles  long, 
24  feet  wide  and  7 feet  deep  in  528  days,  if  42  men,  working  10  hours  a day,  can 
dig  a canal  1J  miles  long,  26  feet  wide  and  6 feet  deep  in  610  days? 

18.  If  45  tons  of  hay  will  be  sufficient  for  S5  horses  96  days,  how  many  tons 
must  be  added  to  be  sufficient  for  125  horses  112  days? 

19.  If  3 men  or  5 boys  can  bind  5 acres  of  oats  in  2J  days,  how  many  acres 
of  the  same  average  yield  could  5 men  and  7 boys  bind  in  3f  days  ? 


ALLIGATION 


828.  Alligation,  or  medial  proportion,  is  the  process  of  combining  two 
or  more  quantities  of  different  values  so  as  to  make  a combination  of  a given 
mean  value. 

829.  To  find  the  average  value  when  the  values  of  the  several 
ingredients  are  given. 


Example. — A grocer  mixes  45  pounds  of  tea  worth  50  cents  a pound,  85 
pounds  worth  60  cents,  125  pounds  worth  75  cents,  and  90  pounds  worth  81- 
Find  the  average  price. 


50  cents  X 45=822.50 
60  cents  X 85=  51.00 
75  centsXl25=  93.75 
$1.00  X 90=  90.00 
$257.25 


value  of  the  45  lbs. 

“ « “ 85  • “ 

“ “ « 125  “ 

“ “ “ 90  “ 

345 


$257.25  -4-  345  = 74^-f  cents,  value  of  1 lb. 


830.  To  find  the  amount  of  the  several  ingredients  taken  when 
their  respective  values  and  the  mean  or  average  value  are  given. 

Example. — It  is  required  to  know  how  many  pounds  of  coffee  at  the  follow- 
ing prices,  25  cents,  32  cents,  38  cents,  40  cents  and  45  cents,  may  be  taken  to 
form  a mixture  worth  35  cents  a pound. 


1 lb.  at  25  cents 

6 “ “ 32  “ 

1 “ “ 38  “ 

3 » « 40  « 

1 “ “ 45  “ 


)■  Answers. 


Explanation, — Arrange  the  prices  of  the  different  ingredients  in  a vertical  column.  Then  link 
one  that  is  above  the  average  price  with  one  that  is  below  it.  In  the  operation  given  above,  we  first 
linked  25  cents  with  45  cents.  Now,  if  one  pound  at  25  cents  be  mixed  and  is  then  worth  35 
cents,  there  is  a gain  of  10  cents,  or  to  gain  one  cent  we  take  -fe  of  a pound  ; and  if  one  pound  at  45 
cents  be  mixed  so  that  it  is  worth  35  cents,  we  lose  10  cents,  or  to  lose  one  cent,  we  take  of  a pound. 
The  loss  of  one  cent  on  T\j  of  a pound  at  45  cents  is  balanced  by  the  gain  of  one  cent  on  rq  of  a 
pound  at  25  cents.  In  the  same  way  we  reason  with  each  of  the  other  combinations,  and  find  next  that 
the  loss  of  one  cent  on  1 of  a pound  at  40  cents  is  offset  by  the  gain  of  one  cent  on  1 of  a pound  at 
32  cents,  and  that  the  loss  of  one  cent  on  J of  a pound  at  38  cents  is  offset  by  a gain  of  one  cent  on  1 of 
a pound  at  32  cents.  Reducing  these  several  sets  of  results  to  whole  numbers  and  carrying  out  the 
results,  we  have  the  answers  given  above. 


253 


254 


ALLIGATION 


Integral  Method 


g.  and  1. 
on  1 lb. 
of 

1 and  5 

g.  and  1. 
on  1 lb. 
of 

2 and  4 

g.  and  1. 
on  1 lb. 
of 

2 and  3 

Amounts 
of  1 and  3 
g.  and  1. 
equal 

Amounts 
of  2 and  4 
g.  and  1. 
equal 

Amounts 
of  2 and  3 
g.  and  1. 
equal 

Amount 

of 

each 

Pboof. 

25 

10 

1 lb. 

1 lb. 

@ 

25c.  = 

§0.25 

32 

3 

3 

5 lbs. 

1 lb. 

6 lbs. 

@ 

32c.  = 

1.92 

38 

3 

1 lb. 

1 lb. 

(& 

38c.  = 

.38 

40 

5 

3 lbs. 

3 lbs. 

@ 

40c.  = 

1.20 

45 

10 

1 lb. 

1 lb. 

@ 

45c.  ■ 

.45 

12  lbs.  )§4.20 

Average  price,  §0.35 


Explanation. — Arrange  the  prices  and  link  as  before.  Now,  if  one  pound  at  25  cents  is  mixed 
so  as  to  be  worth  35  cents,  there  is  a gain  of  10  cents,  and  if  one  pound  at  45  cents  is  mixed  so  as  to  be 
worth  35  cents,  there  is  a loss  of  10  cents  ; again,  if  one  pound  at  32  cents  is  mixed  so  as  to  be  worth 
35  cents,  there  is  a gain  of  3 cents,  and  if  one  pound  worth  40  cents  is  mixed  so  as  to  be  worth  35  cents, 
there  is  a loss  of  5 cents  ; likewise,  if  a pound  worth  38  cents  is  mixed  so  as  to  be  worth  35  cents,  there 
is  a loss  of  3 cents,  and  if  a pound  worth  32  cents  is  mixed  so  as  to  be  worth  35  cents,  there  is  a gain  of 
3 cents.  In  order  to  have  the  gain  and  loss  equal,  one  pound  of  the  25-cent  kind  (gain  10  cents)  must 
be  taken  as  often  as  one  pound  of  the  45-cent  kind  (loss  10  cents) ; 5 pounds  of  the  32-cent  kind  (gain 
15  cents)  as  often  as  3 pounds  of  the  40-cent  kind  (loss  15  cents)  ; and  one  pound  of  the  38-cent  kind 
(loss  3 cents)  as  often  as  one  pound  of  the  32-cent  kind.  Carrying  out  the  results  we  have  the  answers 
as  above. 

831.  To  find  the  number  of  each  ingredient  when  one  or  more  of  the 
ingredients  are  limited. 

Example. — It  is  required  to  know  how  many  gallons  of  alcohol  65%  proof 
and  how  many  gallons  of  water  must  be  mixed  with  90  gallons  of  alcohol  80% 
proof  to  produce  alcohol  60%  proof. 


r 65 

x 

12 

12  gals.,  65% 

80 

) 

i 

2 0 

3x30  = 90 

90 

^00 

) 

¥ (T 

1 

To 

1 

1X30  = 30 

31  gals,  water 

Explanation. — Proceed  as  in  Art.  830,  and  multiply  the  number  obtained  of  the  limited  quan- 
tity by  such  a number  as  will  produce  the  required  amount  of  that  quantity.  Multiply  also  by  the 
same  number  the  quantity  obtained  of  the  kind  to  which  it  was  linked,  and  carry  out  the  results. 


ALLIGATION 


255 


832.  To  find  the  number  of  pounds  of  each  ingredient  when  the 
total  amount  is  limited. 

Example. — A grocer  wishes  to  fill  an  order  for  680  pounds  of  tea  which 
shall  cost  him  55  cents  a pound.  To  produce  this  grade  of  tea,  he  blends  the 


following  grades:  40  cent,  50  cent,  75 
pounds  of  each  does  he  take  ? 


cent,  90  cent  and  $1.00.  How  many 

3X40  = 120  at  40c. 

7 4 11X40  = 410  at  50c. 

1 1X40  = 40  at  75c. 

1 1X40=  40  at  90c. 

1X40  = 40  at  $1  00 

17  17)680(40 

68 

0 


Integral  Method 


g.  and  1. 
on  1 lb. 
of 

1 and  5 

g.  and  1. 
on  1 lb. 
of 

2 and  4 

g.  and  1 
on  1 lb. 
of 

2 and  3 

Amt.  of 
1 and  5 
g.  and  1. 
equal 

Amt.  of  Amt.  of 

2 and  4 2 and  3 

g.  and  1.  i g.  and  1. 
equal  eqnal 

Relative 

Amt. 

of 

each 

Average 

of 

lbs. 

Proof. 

40 

15  c. 

3 lbs. 

3 

120 

@40c.=  4800 

50 

5 

5 

7 lbs.  41bs. 

11 

440 

@ 50c. =22000 

55  ■ 

75 

20 

1 lb. 

1 

>40X  = 

40 

@75c.=  3000 

90 

35 

1 lb. 

1 

40 

@ 90c. = 3600 

1.00 

45  c. 

1 lb. 

1 

40 

@1.00=  4000 



17  X 40  = 680  ) 37400 


55c. 


Explanation. — Proceed  as  iu  Art.  830.  Divide  the  number  of  pounds  required  by  the  tota 
number  obtained  by  comparison,  and  multiply  each  quantity  by  this  number. 


WRITTEN  PROBLEMS 

833.  1.  What  is  the  average  price  of  the  following  mixture  of  tea:  95 
pounds  at  38  cents,  150  pounds  at  45  cents,  68  pounds  at  95  cents  and  75  pounds 
at  $1.00  ? 

2.  In  a certain  school  there  are  150  children  6 years  of  age ; 105  children 
7 years  of  age;  96  children  9 years  of  age  ; 168  children  11  years  of  age,  and  214 
children  12  years  of  age.  Find  the  average  age  of  the  children  of  the  school. 


256 


ALLIGATION 


3.  The  average  price  of  a certain  blend  of  tea,  consisting  of  teas  worth  50, 
60,  65,  75,  80,  85,  90,  95  cents  and  $1,  is  80  cents.  Find  the  number  of  pounds 
of  each  kind  required  to  produce  this  grade  of  tea. 

If.  A broker  bought  221  shares  of  stock  (par  $50)  at  an  average  premium 
of  10%  ; he  bought  some  at  6%  discount,  some  at  2%  discount,  some  at  a pre- 
mium of  16%  and  some  at  a premium  of  20%.  How  many  shares  of  each  kind 
did  he  buy  ? 

5.  A merchant  mixed  95  pounds  of  coffee  worth  35  cents  with  coffees  worth 
40,  42  and  48  cents  a pound.  How  many  pounds  of  each  did  he  take,  if  the 
average  price  was  45  cents  a pound? 

6.  What  relative  quantities  of  alcohol  62%,  68%,  89%,  92%,  96%  and 
98%  proof,  are  required  to  form  alcohol  85%  proof? 

7.  A grocer  has  an  order  for  900  pounds  of  tea,  the  price  to  be  80  cents  a 
pound.  To  fill  the  order  he  mixes  the  following  kinds:  100  pounds  at 42  cents, 
some  at  50  cents,  some  at  90  cents  and  some  at  $1.  How  many  pounds  of  each 
kind  does  he  take  ? 

8.  I want  to  fill  an  order  for  1000  pounds  of  coffee  which  I have  agreed  to 
sell  at  66  cents  a pound,  a gain  of  10%.  I use  200  pounds  at  42  cents,  some 
at  55  cents,  some  at  70  cents  and  some  at  80  cents.  Find  the  number  of  pounds  ■ 
of  each  kind. 

9.  I want  to  mix  wine  costing  $2.50  a gallon  with  90  gallons  of  wine  cost- 
ing $3.00  a gallon  and  with  water,  so  that  I may  be  able  to  sell  the  mixture  for 
$2.40  a gallon  and  gain  20%.  How  man}^  gallons  of  water  and  how  many 
gallons  of  the  wine  at  $2.50  must  I take  ? 

10.  A feed  merchant  mixes  bran  worth  95  cents  a hundred,  middlings 
worth  $1.05  a hundred,  cracked  corn  worth  82  cents  a hundred,  with  oats  worth 
62  cents  a bushel  of  32  pounds.  How  many  pounds  of  each  of  these  ingredients 
shall  he  take  to  fill  an  order  for  3 tons  (2000  pounds  each)  at  an  average  price  of 
$20  a ton  ? 

11.  I want  to  mix  coffees  worth,  respectively,  28,  32,  35,  38  and  40  cents  a 
pound  so  as  to  be  able  to  sell  for  40  cents  a pound  and  gain  10%.  How  many 
pounds  of  each  kind  must  I take? 

12.  I have  on  hand  teas  worth  65,  70,  75,  80  cents,  and  $1  a pound.  I have 
an  order  for  2500  pounds  at  72  cents  a pound.  I use  250  pounds  of  the  $1  tea. 
How  many  pounds  of  the  remaining  kinds  shall  I take? 

13.  A man  has  $190  in  10-cent,  pieces,  which  he  wishes  to  exchange  for  2- 
cent,  5-cent,  25-cent  and  50-cent  pieces.  How  many  of  each  kind  will  it  take  ? 

Ilf  I want  to  mix  alcohol  98%  proof,  95%  proof  and  80%  proof  with  water 
so  as  to  get  alcohol  75%  proof.  Find  the  number  of  gallons  of  each  kind  I must 
take,  if  in  all  I want  1000  gallons  ? 

15.  From  teas  worth  48,  54,  72,  80  and  85  cents  a pound  I want  to  mix  1200 
pounds  of  tea  which  I can  sell  at  80  cents  a joound  and  gain  20%.  By  taking  150 
pounds  of  the  85-cent  tea,  how  many  pounds  of  the  remaining  kinds  must  I 
take  ? 


GENERAL  AVERAGE 


834.  Average  is  of  two  kinds — general  average  and  particular  average. 

835.  General  average  arises  when  sacrifices  have  been  voluntarily  made, 
or  expenditures  incurred,  for  the  preservation  of  a ship,  cargo  and  freight,  from 
some  peril  of  the  sea  or  from  its  effects.  It  implies  a subsequent  contribution 
from  all  the  parties  concerned,  ratable  to  the  values  of  their  respective  interests, 
to  make  good  the  loss  thus  occasioned. 

836.  Particular  average  signifies  the  damage  or  partial  loss  happening 
to  the  ship,  goods  or  freight,  by  some  fortuitous  or  unavoidable  accident.  It  is 
shared  by  the  persons  whose  property  is  destroyed  or  by  their  insurers. 

837.  The  fundamental  principle  of  general  average  is  that  a loss  incurred 
for  the  advantage  of  all  the  coadventurers  should  be  made  good  by  all  in  equitable 
proportion  to  their  stakes  in  the  venture. 

838.  All  general  average  losses  may  be  divided  into  two  principal  classes — 
{1)  sacrifices  of  part  of  the  cargo  and  freight  (as  when  part  of  a cargo  is  thrown 
overboard  to  save  the  ship  from  foundering  in  a storm),  or  of  part  of  the  ship 
(masts,  etc.)  for  the  general  safety;  (2)  extraordinary  expenditures  incurred  with 
the  same  object  (as  when  a ship  is  obliged  to  put  into  a port  of  refuge  in  conse- 
quence of  damage  received  in  the  course  of  the  voyage). 

839.  Jettison  is  the  throwing  overboard  of  goods  or  cargo,  in  stress  of 
weather  or  to  prevent  foundering. 

840.  Salvage  is  the  compensation  allowed  to  person*  by  whose  voluntary 
exertions  a vessel,  her  cargo,  or  the  lives  of  those  belonging  to  her  are  saved 
from  danger  or  loss  in  case  of  wreck,  capture,  or  other  marine  misadventure. 

841.  A general  average  loss  may  include  the  following  items : 

(a)  Jettison:  damage  to  cargo  by  water  getting  down  the  hatches  during 
jettison;  damage  by  breaking  or  chafing  after  jettison;  freight  on  cargo  jetti- 
soned. 

( b ) Sacrifices  of  ship’s  materials,  by  the  cutting  away  of  masts,  spars,  rigging, 
etc.  One-third  of  the  cost  of  repairs  to  a vessel  is  a special  charge  on  the  ship, 
the  new  work  being  considered  better  than  the  old  ; this  leaves  the  remaining 
two-thirds  only  of  such  cost  to  be  included  in  the  general  average. 

(c)  Expense  of  floating  a stranded  ship  ; salvage  in  general. 

(d)  Expense  of  entering  a port  of  refuge,  whether  disability  were  caused  by 
accident  or  sacrifice. 

(e)  Expense  of  discharging  cargo  to  make  repairs,  reloading,  etc. 

(f)  Wages  and  provisions  of  crew  from  time  vessel  deviates  from  its  course 
until  it  resumes  its  voyage. 


257 


258 


GENERAL  AVERAGE 


842.  The  contributory  interests  are  as  follows: 

(a)  The  ship  contributes  on  what  was  its  full  value  before  the  loss. 

(b)  The  cargo  contributes  on  its  net  market  value  at  port  of  destination, 
including  value  of  part  jettisoned  or  damaged. 

(c)  The  freight  contributes  on  its  full  amount,  less  £ for  the  wages,  etc.,  of 
the  crew.  (In  New  York,  and  some  other  States,  less  J). 

843.  Insurers  indemnify  the  owners  of  contributory  interests  for  such 
proportion  of  the  contribution  of  each  as  the  amount  insured  on  such  contrib- 
utory interest  bears  to  the  full  value  of  said  contributory  interest. 

844.  An  adjuster  is  a person  who  apportions  the  losses  and  expenses  of  a 
general  average. 

845.  To  apportion  a loss  by  general  average. 

Example. — The  schooner  Swallow  from  Savannah  for  Philadelphia,  with 
a cargo  of  cotton  valued  at  $37800  and  a deck  load  of  pine  lumber  valued  at 
$18760,  encountered  heavy  gales  off  Cape  Hatteras  which  made  it  necessary  to 
throw  overboard  the  deck  load  and  cut  away  masts  and  rigging,  after  which  the 
vessel  was  rescued  by  a steamer  and  towed  into  Wilmington,  N.  C.  The  salvage 
amounted  to  $15000  ; repairs  to  the  vessel,  $2871 ; wages  and  provisions  of 
seamen  from  date  of  the  disaster,  $874.60  ; the  other  expenses,  $1486.38.  The 
value  of  the  vessel  was  $12000  ; the  total  freight,  $6438,  of  which  $2132  was  the 
freight  on  the  deck  load.  Apportion  the  settlement. 


GENERAL  AVERAGE 

LOSS 

CONTRIBUTORY  INTEREST 

Jettison 

18760 

Vessel 

12000 

Freight  on  jettison 

2132 

Cargo  of  cotton 

37800 

Repairs  (f  of  $2871) 

1914 

Cargo  of  lumber 

18760 

Salvage 

15000 

Freight  ($6438  less  ^) 

4292 

Wages,  etc.,  of  seamen 

874 

60 

Other  expenses 

1486 

38 

Total 

40166 

98 

Total 

72852 

$72852  : $40166.98  : : $12000  : Vessel’s  contribution  = $6616.20 

$72852  : $40166.98  : : $37800  : Cotton’s  contribution  = 20841.05 

$72852  : $40166.98  : : $18760  : Lumber’s  contribution  = 10343.33 

$72852  : $10166.98  : : $4292  : Freight’s  contribution  = 2366.40 

$40166.98 


(receives)  Adjuster’s  Settlement  (pays) 


Owners  of  cotton 

20841 

05 

Owners  of  lumber 

8416  67 

Freight  company 

234 

40 

Repairs 

2871 

Owners  of  vessel 

7573 

20 

Salvage 

15000 

Wages,  etc.,  of  seamen 

874  60 

Other  expenses 

I486  38 

2864S 

65 

28648  65  j 

GENERAL  AVERAGE 


259 


Explanation  of  Adjuster’s  Settlement. — The  adjuster  settles  with  the  owners  of  the  cotton 
by  receiving  from  them  their  contribution  to  the  general  average,  $20841.05.  He  settles  with  the  freight 
company  (the  charterers  of  the  vessel)  by  receiving  from  them  the  difference  between  their  contribu- 
tion and  the  freight  on  jettison,  $2366.40 — $2132  = $234.40.  He  settles  with  the  owners  of  the  vessel 
by  receiving  from  them  their  contribution  plus  their  £ of  the  repairs,  $6616.20  -j-  $957  =$7573.20.  He 
settles  with  the  owners  of  the  lumber  by  paying  them  the  value  of  the  lumber  jettisoned,  less  their 
contribution,  $18760 — $10343  33  = $8416.67.  He  pays  the  full  amount  for  repairs,  $2871  ; for  salvage, 
$15000  ; and  other  expenses,  $1486.38.  He  pays  the  general  average  portion  of  seamen’s  wages,  etc., 
the  charterers  paying  their  regular  wages.  Of  course,  the  charterers  receive  their  freight  charges  from 
the  owners  of  the  cotton,  and  they  pay  the  owners  of  the  vessel  for  the  charter  of  it,  entirely  aside  from 
the  general  average  settlement. 

846.  Rule. — Find  the  total  of  the  contributory  interests. 

Find  the  total  loss  subject  to  general  average. 

Calculate  the  proportionate  contribution  of  each  contributory  interest  by  the 
following  formula : 

Total  interest : Total  loss  : : Each  owner's  interest : Each  oivner's  contribution. 

WRITTEN  PROBLEMS 

847.  l.  A steamer  having  on  board  $47923  worth  of  goods  shipped  by  A, 
$25437  worth  shipped  by  B,  and  $11926  worth  shipped  by  C,  experienced  such 
rough  weather  that  $20624  worth  of  goods  were  jettisoned  for  the  safety  of  the 
ship.  Of  the  goods  jettisoned,  $8370  worth  were  from  goods  owned  by  A,  and 
the  remainder  from  B’s  shipment.  The  vessel  having  been  damaged,  the  repairs 
amounted  to  $2145.38,  and  the  maintenance  of  the  crew  during  delay  was 
$537.80.  The  loss  of  freight  on  cargo  jettisoned  amounted  to  $2085.50  ; the 
freight  paid  on  the  remainder  was  $6249.80.  The  steamer,  before  starting  on 
the  voyage,  was  valued  at  $57500.  Apportion  the  loss  among  the  different 
contributory  interests,  and  state  how  settlement  should  be  made. 

£.  The  bark  Bonnie  Jean  started  from  Glasgow  to  New  York  with  an 
assorted  cargo.  During  the  voyage  she  encountered  a severe  storm,  driving  her 
out  of  her  course  and  causing  considerable  damage.  For  the  safety  of  the  ship 
her  masts  and  rigging  had  to  be  cast  away  ; also  a portion  of  her  cargo,  valued 
at  $2375.  Bv  rigging  up  jury  masts,  she  was  enabled  to  reach  the  port  of  St. 
John,  N.  B.,  for  repairs.  Replacing  the  masts  and  rigging  cost  $5438  ; repairing 
other  damage  $675.  The  freight  on  cargo  jettisoned  amounted  to  $435.25. 
Port  charges  for  entering,  with  expense  of  discharging  cargo,  etc.,  amounted  to 
$835.15  ; wages  and  provisions  for  crew,  $715  ; adjuster’s  fee,  $175.  The  value  of 
the  vessel  on  arriving  at  New  York  was  $28000 ; value  of  cargo  delivered,  $50860. 
The  total  freight  (including  freight  on  cargo  jettisoned)  was  $17150.  The  con- 
signors of  the  cargo  were  : James  Thompson,  $9500  ; William  Irvine,  $21275  ; 
Alexander  McPherson,  $7325 ; Joseph  Montgomery,  $15135.  The  jettisoned 
goods  were  part  of  Irvine’s  consignment.  How  ought  the  settlement  to  be 
made  ? 

Note. — The  value  of  the  vessel  on  arriving  at  New  York  being  $28000,  to  find  the  value  before 
it  was  damaged  (which  will  be  the  value  upon  which  its  contribution  is  computed)  deduct  one  third  of 
the  charges  for  repairs,  this  being  reckoned  as  the  superior  value  of  the  new  material.  That  is  to  say, 
the  vessel,  after  being  repaired,  is  worth  that  much  more  than  before  the  damage  was  sustained. 


260 


GENERAL  AVERAGE 


3.  James  Thompson’s  consignment  was  insured  for  $8000,  and  Joseph 
Montgomery’s  consignment  for  $12000.  How  much  did  each  receive  from  the 
underwriters? 

J.  During  a storm  off  the  cost  of  Maine,  the  schooner  Sarah  Jane , valued 
at  $25000,  having  a cargo  consigned  to  Frazer  & Co.,  Baltimore,  had  to  jettison 
$5000  worth  of  the  cargo  for  the  general  safety.  The  cargo  amounted  to  $50500, 
part  of  which  was  consigned  by  Jones  & Brown  and  invoiced  at  $15750,  of  which 
$3000  was  jettisoned.  The  remainder  of  the  jettison,  $2000,  was  taken  from 
goods  consigned  by  Bradley  & Co.  By  the  shifting  of  the  cargo,  and  the  enter- 
ing of  water  through  the  hatchway  while  throwing  the  goods  overboard,  the 
remainder  of  Jones  & Brown’s  consignment  was  damaged  to  the  extent  of  1\%. 
Freight  expected  on  cargo  was  $12500,  including  that  on  jettison,  $315.  The 
adjuster’s  fee  was  $175.  How  much  do  Jones  & Brown  and  Bradley  & Co.  each 
contribute  in  the  general  average? 

5.  If  Bradley  & Co.  carried  $30000  insurance  on  their  consignment,  how 
much  do  they  receive  from  the  insurance  company? 

6.  The  brig  Arcotta — owned  by  Jacob  Orr,  Win.  Coates  and  Henry  Fox, 
general  traders,  whose  respective  interests  in  the  vessel  were,  Orr  J,  Coates  f , and 
Fox  J — having  delivered  cargo  at  several  of  the  principal  seaports  of  the  Medi- 
terranean, shipped  a cargo  of  oranges  and  lemons  at  Messina  for  the  return 
voyage,  each  of  the  owners  putting  in  his  own  venture.  The  value  of  the  cargo 
was  $43575,  which  was  as  follows  : Jacob  Orr  $14525,  William  Coates,  $7262  50. 
Henry  Fox  $9078.13,  and  a consignment  by  Wessenberg&  Co.  to  Penn  Fruit  Co., 
Philadelphia,  $12709.37.  On  the  homeward  voyage  the  brig  was  struck  by  a 
violent  storm,  during  which  the  vessel  was  damaged  to  a considerable  extent, 
Some  of  the  cargo  had  to  be  thrown  overboard,  and  having  sprung  a leak,  the 
vessel  had  to  put  into  Bordeaux  for  repairs.  Expenses  for  repairing  loss  and 
damages,  port  charges  for  entering,  wages  and  provisions  for  crew  while  detained, 
unloading,  reloading,  etc.,  amounted  to  7532.75  francs,  which  was  paid  by  the 
owners,  Wessenberg’s  share  to  be  collected  from  the  Penn  Fruit  Co.  Of  the 
total  expenses,  1236.20  francs  was  the  amount  paid  for  repairing  damage  to 
ve-sel  caused  by  the  storm.*  The  goods  jettisoned  were  valued  at  $8500 — $4500 
being  from  J.  Orr’s  consignment  and  $4000  from  Wessenberg’s  consignment. 
Freight  calculated  at  $8400,  including  $345  freight  on  jettison.  Value  of  brig  in 
Philadelphia,  $25000.  Adjuster’s  fee,  $225.  How  was  settlement  made? 

7.  The  cargo  of  the  brig  Arcotta  was  insured  for  $38000.  How  much  did 
the  insurance  company  pay,  and  how  apportioned? 


* This  item  is,  of  course,  a particular  average  loss  to  be  shared  by  the  owners. 


EQUATION  OF  ACCOUNTS 

848.  Equating  an  account  is  finding  a date  on  which  settlement  of  two 
or  more  debts,  due  at  different  dates,  can  be  made  without  loss  to  the  debtor  or 
creditor. 

849.  The  date  thus  found  is  called  the  average  date  or  equated  time. 

850.  It  is  necessary  to  assume  some  common  date  of  comparison.  This  date 

is  called  the  focal  date. 

Remark. — Any  date  conceivable  may  be  taken  as  a focal  date,  and  interest  may  be  computed  at 
any  rate  per  cent,  without  varying  the  result ; providing  only  that  the  dates  of  all  items  be  compared 
with  such  focal  date,  and  uniformity  in  rate  and  manner  of  computing  interest  be  observed  throughout. 

In  practise  it  is  well  to  observe  a simple  method,  by  assuming  the  latest  date  in  the  account  as 
a focal  date,  computing  all  interest  at  6%  by  the  short  method  on  a 360-day  basis. 

851.  The  term  of  credit  is  the  time  allowed  for  the  payment  of  a debt  after 
the  date  of  contracting  it ; if  given  in  days,  it  is  counted  on  from  the  date  of  pur- 
chase or  sale,  the  exact  number  of  days  of  the  term;  if  given  in  months,  it  is 
counted  on  the  number  of  months  regardless  of  the  number  of  days  thus  included. 

852.  The  average  term  of  credit  is  the  average  time  allowed  for  the  pay- 
ment, in  one  sum,  of  the  total  of  two  or  more  debts,  due  at  different  dates. 

Remark. — Book  accounts  bear  legal  interest  after  they  become  due,  and  notes,  even  if  not  con- 
taining an  interest  clause,  bear  interest  after  maturity. 

853.  The  importance  of  a thorough  knowledge  of  both  the  theory  and  prac- 
tise of  Equation  of  Accounts,  on  the  part  of  bookkeepers  and  accountants,  can 
hardly  be  overrated,  as  much  of  this  class  of  work  is  to  be  found  in  every  whole- 
sale and  commission  business. 

854.  The  equity  of  the  settlement  of  an  account  by  equation  rests  in  the 
fact  that,  by  a review  of  such  account,  one  of  the  parties  owes  the  other  a balance 
to  which  certain  interest  should  be  added  or  from  which  certain  interest  (dis- 
count) should  be  subtracted. 

855.  Accounts  having  entries  on  but  one  side,  either  debit  or  credit,,  are 
appropriately  called  simple  accounts;  accounts  having  both  debit  and  credit 
items  may  appropriately  be  called  compound  accounts. 

856.  It  is  the  custom  in  many  lines  of  business  to  charge  interest  on  items 
that  have  matured.  Brokers  also  charge  customers  interest  for  stock  carried  or 
allow  it  on  deposits  made.  Private  and  state  banks  frequently  allow  interest  on 
daily  balances  on  deposit.  These  various  business  customs  lead  to  interest  cal- 
culations of  special  kinds,  with  which  the  business  man  should  be  familiar. 
There  are  several  methods  in  use. 


261 


262 


EQUATION  OF  ACCOUNTS 


857.  To  find  the  average  date  of  sales. 

Example. — In  the  following  account  find  the  average  date  of  sales:  on 
June  8,  1907,  John  Bowen  sold  to  William  Lorigan  mdse,  to  the  amount  of  §960  ; 
on  June  16,  to  the  amount  of  $1240  ; and  on  June  30  to  the  amount  of  $497. 


r 

/& 

30 


*9ru^ 


2 / / 20 

/ 73  6>0 

0 oooo 

J 3 & 4^ '<f  0 ( / 

- • 

/ / S / 0 
/ oy 
yz  x 


Explanation. — We  assume  the  latest  date  of  sale,  June  30,  as  the  time  of  settlement.  If  the 
first  item  is  paid  on  that  date,  it  bears  interest  for  a period  of  22  days,  or  $1  has  an  interest  period  of 
21120  days.  The  second  item,  purchased  June  16,  bears  interest  for  a period  of  14  days,  or  $1  has  an 
interest  period  of  17360  days.  If  all  items  are  paid  June  30,  $1  has  an  interest  period  of  38480  days,  or 
$2697  for  14  days.  Since  Mr.  Lorigan,  by  paying  June  30,  owes  the  interest  on  $2697  for  14  days,  it  is 
evident  that  the  day  on  which  he  owes  no  interest  must  be  14  days  earlier  than  June  30,  or  June  16. 

858.  Rule. — Assume  as  a focal  date  the  latest  date  of  sale  in  the  account.  Mul- 
tiply each  item  in  the  account  by  the  number  of  days  between  the  date  of  sale  and  the 
focal  date.  The  sum  of  these  products  divided  by  the  sum  of  the  items  will  give  the 
average  credit  before  the  focal  date. 

Note. — A fraction  of  a day,  if  less  than  one-half,  is  discarded  ; if  one-half  or  more,  it  is  called  a 
day.  If  an  item  contains  cents,  they  maybe  discarded  if  less  than  one-half  dollar.  One  half  dollar  or 
more  may  be  called  one  dollar. 

859  To  find  the  average  due  date. 


Example. — In  the  following  statement  find  the  average  due  date: 


1906 

Mar. 

7 

Mdse. 

30 

ds. 

200 

40 

May 

9 

U 

2 

Ill  os. 

375 

00 

Aug. 

17 

it 

60 

ds. 

400 

60 

EQUATION  OF  ACCOUNTS 


263 


Product  Method 


1906 

DUE 

ITEMS 

DS. 

PRODUCTS 

Mar. 

7 

Mdse.  30  ds. 

April  6 

200 

40 

ZD 

CO 

38600 

May 

9 

“ 2 mos. 

July  9 

375 

00 

99 

37125 

Aug. 

17 

“ 60  ds. 

Oct.  16 

400 

976 

60 

0 

00000 

976)75725(77  + 1=78  ds. 

78 

16  Oct. 
62 

30  Sept. 
32 


32 

31  _Aug. 

1 31  July 

1 

30  “ 


6832 

7405 

6832 

573 

Average  due  date,  July  30. 


Explanation. — First  find  the  due  date  of  each  item.  Assume  October  16,  the  latest  due  date, 
■as  the  date  of  settlement  for  all  the  items.  If  the  first  item  is  paid  on  that  date,  it  bears  interest  for  a 
period  of  193  days  ; or  $1  has  an  interest  period  of  38600  days. 

The  second  item,  due  July  9,  has  an  interest  period  of  99  days  ; or  $1  has  au  interest  period  of 
37125  days.  If  all  items  are  paid  October  16,  $1  has  an  interest  period  of  75725  days,  or  §976  for  78 
-days.  Since  the  purchaser  owes  the  interest  on  §976  for  78  days  by  paying  October  16,  not  to  owe  this 
interest  he  must  pay  78  days  before  October  16,  or  July  30. 


860.  Rule — Assume  the  latest  due  date  of  the  account  as  a focal  date.  Multiply 
■each  item  in  the  account  by  the  number  of  days  between  the  due  date  of  the  item  and 
this  focal  date.  The  sum  of  the  products  divided  by  the  sum  of  the  items  gives  the 
average  credit  (in  days ) before  the  focal  date. 

861.  V erification  by  using  earliest  due  date  as  the  focal  date. 


1906 

DUE 

ITEMS 

DS. 

PRODUCTS 

Mar. 

7 

Mdse. 

30  ds. 

April  6 

200 

40 

0 

00000 

May 

9 

U 

2 mos. 

July  9 

375 

00 

94 

35250 

Aug. 

17 

u 

60  ds. 

Oct.  16 

400 

60 

193 

77393 

115 

976 

976)112643(115 

976 

24  April  1504 

91  976 


31  May 
60 

30  June 

30  July,  average  due  date 


5283 

4880 

403 


Explanation. — Assume  April  6,  the  earliest  due  date,  as  the  focal  date  for  all  the  items.  If 
the  first  item  is  paid  on  that  date,  it  neither  bears  interest  nor  is  it  entitled  to  a discount.  If  the 
second  item  (due  July  9)  is  paid  April  6,  it  is  entitled  to  a discount  period  of  94  days,  or  §1  is  entitled 
to  a discount  period  of  35250  days.  The  third  item  (due  October  16)  is  entitled  to  a discount  period 
of  193  days,  or  §1  is  entitled  to  a discount  period  of  77393  days.  If  all  items  are  paid  on  April  6,  §1 
has  a discount  period  of  112643  days,  or  §976  for  115  days.  Since  the  purchaser  is  entitled  to  the  dis- 
count on  §976  for  115  days,  not  to  be  entitled  to  this  discount  he  should  pay  115  days  after  April  6,  or 
July  30. 


264 


EQUATION  OF  ACCOUNTS 


862.  Rule. — Assume  as  a focal  date  the  earliest  due  date  of  the  account.  Mul- 
tiply each  item  in  the  account  hy  the  number  of  days  between  this  focal  date  and.  the 
due  date  of  the  item.  The  sum  of  the  products  divided  by  the  sum  of  the  items  gives 
the  average  {in  days ) after  the  focal  date. 


Interest  Method 


1906 

DUE 

DS. 

ITEMS 

INTEREST 

Mar. 

7 

Mdse.  30  ds. 

April  6 

193 

200 

40 

$6 

4462 

May 

9 

“ 2 mos. 

July  9 

99 

375 

00 

6 

1875 

Aug. 

17 

“ 60  ds. 

Oct.  16 

0 

400 

976 

60 

00 

0 

)12 

0000 

6337(77  + l=7S  ds. 

•000L 

.16266  ) 12.6337  ( 77  16266 

11  3862 
1 24750 

1 13862  Average  due  date,  July  30. 

Explanation.- — Assume  as  a date  of  settlement  the  latest  due  date,  October  16.  If  he  did 
not  pay  $200.40  (due  April  6)  until  October  16,  he  owes  the  interest  on  that  amount  for  193  days,  or 
$6.4462.  In  like  manner  $375  (due  July  9)  was  paid  October  16,  or  99  days  later  ; therefore,  the 
interest  due  is  $6.1875.  All  the  interest  (due  October  16)  amounts  to  $12.6337.  Not  to  owe  this 
interest  October  16,  he  should  have  paid  as  many  days  earlier  as  the  interest  on  the  total  for  one  day 
($0.16266)  is  contained  times  into  $12.6337,  or  78.  78  days  earlier  than  October  16,  1906,  equals  July  30. 

863.  To  find  the  equated  date  for  the  payment  of  an  account. 

Example. — In  the  following  statement  find  the  time  when  the  balance  is 
due  ; or,  “ equate  ” the  following  account : 


Dr.  John  K.  Yakdham  C’r. 


1907 

190 

7 

Feb. 

15 

To  Mdse. 

600 

Mar. 

10 

By  Cash 

350 

May 

25 

it  it 

400 

June 

12 

tt  tc 

300 

Dr.  Product  Method  Cr. 


1907 

DS. 

ITEMS 

PRO. 

19u 

7 

DS. 

ITEMS 

PRO. 

Feb. 

May 

15 

25 

To  Mdse. 

tt  tt 

117 

18 

600 

400 

1000 

00 

00 

70200 
7200  i 
77400 

Mar. 

June 

10 

12 

By  Cash 

tt  tt 

94 

0 

350 

300 

650 

00 

00 

329u0 

00000 

32900 

650  32900 

350)  44500(127 


4200 

127 

84 

2500 

12  June 

30  Apr. 

28  Feb. 

2450 

115 

54 

23 

50 

31  May 

31  Mar. 

5 Feb. 

^84 

23 

equated  date. 

EQUATION  OF  ACCOUNTS 


265 


Explanation. — In  the  above  illustration  we  take  the  latest  date  of  the  account,  June  12,  as 
the  assumed  date  of  settlement. 

Suppose  the  first  item  ($600)  to  have  been  paid  on  this  assumed  date  ; it  would  have  been  paid 
117  days  after  it  was  due,  and  Yardham  would  owe  the  interest  on  $600  for  117  days,  which  is  equal  to 
an  interest  period  on  $1  for  70200  days.  Had  the  second  item  ($400)  been  paid  on  this  assumed  date, 
he  would  owe  interest  on  $400  for  18  days,  which  is  equal  to  an  interest  period  on  $1  for  7200  days. 
Yardham,  therefore,  on  June  12,  owes  $1000  and  the  interest  on  $1  for  a period  of  77400  days. 

On  the  credit  side  of  the  account,  by  using  the  same  date  of  comparison,  since  he  paid  $350  on 
March  10,  or  94  days  before  the  date  of  settlement,  he  is  entitled  to  a discount  on  $350  for  94  days,  or 
on  $1  for  a period  of  32900  days.  The  second  item  was  paid  June  12,  and  is  not  entitled  to  a discount. 

Subtracting  the  number  of  days  for  which  he  is  to  be  allowed  the  discount  on  $1  from  the  num- 
ber of  days  for  which  he  owes  the  interest  on  $1,  viz.,  77400 — 32900=44500.  the  number  of  days  for 
which,  on  June  12,  he  owes  the  interest  on  $1.  If  he  owes  the  interest  on  $1  for  44500  days,  he  owes 
the  interest  on  $350,  the  balance  of  the  account,  for  as  many  days  as  $350  is  contained  times  into  44500, 
or  127.  If,  on  June  12,  1907,  he  owes  the  interest  on  the  balance  for  127  days,  this  balance  was  due 
127  days  before  June  12,  or  February  5. 

864.  Rule. — Assume  the  latest  due  date  in  the  account  as  a focal  date.  Multiply 
each  item  on  both  sides  of  the  account  by  the  number  of  days  between  the  due  date  of 
the  item  and  this  focal  date.  Divide  the  difference  between  the  sums  of  the  products 
by  the  difference  between  the  sums  of  money.  If  these  differences  are  on  the  same  side 
of  the  account,  count  the  number  of  days  obtained  by  the  division  backward  from  the 
focal  date;  if  on  opposite  sides,  count  forward  from  the  focal  date. 

865.  To  verify  the  above  result,  select  a different  date  as  a focal  date. 


Dr.  Cr. 


190 

1 

DS. 

ITEMS 

PRO. 

1907 

DS. 

ITEMS 

PEO. 

Feb. 

15 

To  Mdse. 

0 

600 

00000 

Mar. 

10 

By  Cash 

23 

350 

8050 

A fay 

25 

<t  U 

99 

400 

39600 

June 

12 

u tt 

117 

300 

35100 

1000 

39600 

650 

43150 

650  39600 

~350)3550  3550 

10  days. 

10  days  before  Feb.  15  = Feb.  5 

Explanation. — -Assume  February  15,  1907,  the  earliest  date,  as  the  focal  date.  If,  on  February 
15,  Yardham  pays  $600  (the  value  of  merchandise  bought  on  that  day),  he  pays  his  debt  when  due, 
and  should  neither  be  charged  with  interest  nor  credited  with  discount  ; but  if,  on  February  15,  he 
pays  $400  (not  due  until  May  25),  he  should  be  allowed  a discount  on  that  item  for  99  days  ; or  $1 
has  a discount  period  of  39600  d ays. 

On  the  credit  side  of  the  account,  by  using  the  same  date  of  comparison,  since  he  paid  $350  on 
March  10,  or  23  days  after  the  assumed  date  of  settlement,  he  owes  the  interest  on  that  item  for  23  days  ; 
or  $1  has  an  interest  period  of  8050  days.  The  second  item  ($300)  was  paid  June  12,  or  117  days  after 
the  assumed  date  of  settlement ; hence  he  owes  the  interest  on  that  item  for  117  days,  or  $1  has  an 
1 interest  period  of  35100  days. 

Subtracting  the  number  of  days  for  which  he  is  to  be  allowed  the  discount  on  $1  from  the  num- 
, her  of  days  for  which  he  owes  the  interest  on  $1,  viz.,  43150 — 39600=3550,  the  number  of  days  for 
which,  on  February  15,  he  owes  the  interest  on  $1. 

If  he  owes  the  interest  on  $1  for  3550  days,  he  owes  the  interest  on  $350,  the  balance  of  the 
1 account,  for  as  many  days  as  $350  in  contained  times  into  3550  or  10.  If  on  February  15  he  owes 
the  interest  on  the  balance  for  10  days,  this  balance  is  due  10  days  before  February  15,  or  February  5. 


266 


EQUATION  OF  ACCOUNTS 


866.  Rule. — Assume  the  earliest  due  date  in  the  account  as  a focal  date.  Multi- 
ply each  item  on  both  sides  of  the  account  by  the  number  of  days  between  this  focal 
date  and  the  due  date  of  the  item.  Divide  the  difference  between  the  mms  of  the  pro- 
ducts by  the  difference  between  the  sums  of  money.  If  these  differences  are  on  the  same 
side  of  the  account,  count  the  number  of  days  obtained  by  the  division  forward  from 
the  focal  date  ; if  on  different  sides,  count  backward  from  the  focal  date. 

Proof — Product  Method 

867.  If  Feb.  5,  1907,  is  the  correct  equated  date,  then  the  sum  of  the 
products  obtained  by  multiplying  each  item  on  the  debit  side  by  the  number  of 
days  between  the  equated  date  and  its  due  date  should  equal  the  sum  of  the 
products  obtained  by  multiplying  each  item  on  the  credit  side  by  the  number  of 
days  between  the  equated  date  and  its  date,  to  within  one-half  the  amount  of  the 
unpaid  balance. 


600  X 10  = 6000 

350  X 33  = 11550 

400  X 109  = 43600 

300  X 127  = 38100 

1000  49600 

650  49650 

650 

49600 

350 

50 

The  difference  of  the  sums  of  these  products,  50,  being  less  than  one  half 
of  $350,  the  proof  is  established. 

Note. — If  the  equated  date  falls  between  the  extreme  dates  of  an  account,  that  is,  between  the 
first  and  last  due  dates,  the  difference  between  the  interest  items  and  discount  items  on  the  debit  side 
must  equal  the  difference  between  the  interest  items  and  discount  items  on  the  credit  side  to  within 
one-half  the  balance  of  the  account. 

868.  To  find  the  equated  date  for  the  payment  of  an  account,  using 
as  the  focal  date  the  date  of  the  latest  transaction  or  latest  due  date. 

Example. — In  the  following,  find  the  equated  date: 


Dr. 

J.  H. 

Butler 

Cr. 

1907 

1907 

Ma}r  15 

To  Mdse.,  10  da. 

2S5  00 

June  12 

By  Cash  200  00 

June  18 

“ “ 30  “ 

340  ; 00 

July  1 

Note,  60  da.  300  00 

Dr. 

Interest  Method 

Cr. 

1907 

DUE 

D. 

I NT. 

ITEMS 

1907 

DUE 

D. 

I XT. 

ITEMS 

May 

15 

Mdse.,  10  da. 

May 

25 

97 

4 

6075 

285 

00 

I 

Junejl2 

Cash 

June 

12 

79 

2 6333 

200  00 

June 

18 

“ 30  “ 

July 

18 

43 

2 

4366 

340 

00 

J uly  1 

Note,  60  da. 

Aug. 

30 

0 

0 0000 

300  00 

7 0441  625  00  2 6333  500  00 


EQUATION  OF  ACCOUNTS 


2G7 


212 

7.0441  625.00 

125 

30  Aug. 

2.6333  500.00 

.000J 

182 

90 

4.4100  125.00 

.0208)4.4108(212  days 

31  July 

30  Apr. 

4 16 

151 

60 

250 

30  June 

31  Mar. 

31  Jan. 

208 

121 

29 

1 

428 

31  May 

28  Feb. 

30  “ 

416 

90 

1 

Aug.  30  (back  212  da. ) 

= Jan.  30  Ans. 

Explanation. — Assume  the  latest  due  date,  August  30,  as  the  focal  date.  If  on  August  30 
Butler  is  charged  with  $285  (the  value  of  merchandise  sold  to  him  May  15 and  due  May  25),  he  should 
also  he  charged  with  the  interest  for  97  days  (May  25  to  August  30),  because  he  did  not  pay  for  the 
merchandise  when  it  was  due  ; that  is,  he  should  on  this  item,  be  charged  for  $4.6075  interest.  And 
if,  on  August  30,  he  is  charged  with  $340  (the  value  of  merchandise  sold  to  him  June  18  and  due  July 
18),  he  should  also  be  charged  with  its  interest  for  43  days  (July  18  to  August  30),  because  he  did  not 
pay  for  the  merchandise  when  it  was  due  ; that  is,  he  should  on  this  item  be  charged  $2.4366  interest. 
Thus,  he  is  charged  a total  of  $7.0441  interest,  and  his  total  debt  is  $632.04  on  August  30  provided  he 
had  no  credits  for  payments  prior  to  August  30. 

But  we  have  the  credit  side  of  the  account  to  consider.  If  on  August  30  Butler  receives  credit 
for  $200  paid  on  June  12,  he  should  receive  credit  also  for  the  interest  (discount)  on  the  payment  for  79 
days  between  June  12,  when  he  paid  it,  and  August  30,  the  assumed  date  of  settlement ; that  is,  he 
should  be  credited  for  $2.6333.  If  on  August  30,  he  received  credit  for  $300  paid  on  that  day,  he  should 
not  receive  credit  for  any  interest  (discount),  because  the  money  was  paid  on  the  day  it  fell  due.  If 
there  were  no  debits  or  charges  against  him,  he  would  be  entitled,  on  August  30,  to  a net  credit  of  $500, 
the  cash  paid,  and  $2.6333,  the  interest  (discount);  that  is,  to  $502.63,  which  is  a balance  in  his  favor. 
And  the  cash  balance  due  August  30  would  be  $632.04,  less  $502.63,  which  is  $129.41. 

However,  it  is  not  the  cash  balance  due  August  30,  1907,  that  concerns  us,  but  the  date  on  which 
in  equity  the  balance  of  the  account  ($125)  should  be  paid. 

Since  Butler  was  charged  with  items  amounting  to  $625  and  with  interest  thereon  amounting  to 
$7.0441,  his  debt  on  August  30  would  appear  to  be  the  sum  of  the  two  ; but  not  so.  His  debt  on 
August  30  was  $625  (the  sum  charged)  less  $500  (the  sum  credited)  or  $125  ; but  he  also  owes 
$7.0441  (the  interest  charged)  less  $2.6333  (the  interest  credited)  or  $4.4108. 

For  Butler  not  to  owe  this  interest  August  30,  he  should  have  paid  earlier.  To  determine  the 
number  of  days,  divide  the  balance  of  interest  $4.4108  by  the  interest  on  the  balance  for  1 day,  $.0208, 
and  find  the  time  to  be  212  days.  Since  on  August  30  Butler  owed  not  only  $125  but  also  $4.4108 
interest,  he  had  at  that  date  been  owing  $125  for  a time  (212  days)  sufficient  for  it  to  accumulate 
$4.4108  interest  ; and  if  he  had  on  August  30,  1907,  been  owing  $125  for  212  days,  that  debt  must  have 
been  due  in  equity  212  days  earlier  than  August  30  ; that  is,  January  30,  1907. 

869-  Rule. — Assume  the  latest  due  date  in  the  account  as  a focal  date.  Find 
the  interest  ( to  four  places)  on  each  item  of  the  debit  side,  supposing  each  to  be  paid  on 
this  assumed  date.  Find  the  interest  on  each  item  of  the  credit  side  (to  four  places), 
supposing  each  to  be  due  on  this  assumed  date.  Find  the  difference  between  the  sum  of 
the  debit  interests  and  the  sum  of  the  credit  interests.  Divide  this  difference  by  the 
interest  on  the  bala.nce  of  the  account  for  1 day.  The  quotient  expresses  the  number 
of  days  between  the  assumed  date  and  the  date  on  which  the  balance  of  the  account  is 
due.  If  the  difference  between  the  sums  of  money  and  the  difference  between  the  interests 
are  on  the  same  side,  count  backward ; if  on  opposite  sides,  count  forward  from  the 
focal  date. 


268 


EQUATION  OF  ACCOUNTS 


Remark. — The  date  of  the  latest  transaction  seems  the  more  natural  date  to  use,  and  hence  is 
more  practical  than  assuming  a date  that  does  not  appear  to  exist.  The  number  of  days  is  fewer  and 
there  is  one  less  item  to  calculate.  A great  advantage  in  using  the  interest  method  is  that  it  gives 
excellent  drill  in  calculating  interest ; and  by  the  use  of  the  60-day  method  it  is  as  quickly  worked 
as  by  the  product  method.  It  has  the  advantage  also  of  being  more  easily  explained  and  more  readily 
understood.  It  is  also  noted  that  many  business  houses  use  the  interest  method  exclusively,  in  render- 
ing accounts  current,  especially  in  foreign  trade.  The  form  of  account  current  used  shows  the  items  of 
days  and  interest  in  detail,  having  separate  columns  for  each. 

870.  Verification  by  using  dale  of  the  earliest  transaction  or  the  earliest 
due  date.  Focal  date  May  25,  1907. 

Dr  Ck. 


1907 

DUE 

D. 

INT. 

ITEMS 

1907 

DUE 

D. 

INT. 

ITEMS 

May 

15 

Mdse.,  10  da. 

May 

25 

0 

0 

0000 

285 

00 

June 

12 

Cash 

June 

12 

,8 

6000 

200 

00 

June 

18 

“ 30  “ 

July 

18 

54 

3 

0599 

340 

00 

July 

1 

Note,  60  da. 

Aug. 

30 

97 

4 

8500 

300 

00 

3.0599  625.00  5.4500  500.00 

500.00  3.0599 

125.00  2.3901 


125 

.0001, 


0208)2. 3901  ( 114  days  + 

1 day  or  115  days 

2 08 

25  May 

310 

90 

208 

30  Apr. 

1021 

60 

832 

31  Mar. 

31  Jan 

189 

29 

1 

28  Feb. 

TlO  “ 

May  25  (back  115  days)  = Jan.  30  Ans. 

Explanation.— Select  May  25,  1907,  the  earliest  due  date,  as  the  assumed  date  of  settlement. 
If  on  May  25  Butler  pays  $285  (the  value  of  merchandise  bought  May  15  at  10  days),  he  pays  his  debt 
when  due,  and  should  neither  be  charged  with  interest  nor  credited  with  interest  (discount);  but  if  on 
May  25  he  pays  $340  not  due  until  July  18,  he  should  be  credited  with  discount  on  that  item  for  the 
54  days  between  May  25,  when  he  paid  it,  and  July  18,  when  it  becomes  due  ; that  is,  he  should  be 
credited  with  $3.0599  discount  for  the  prepayment  of  this  item.  Hence,  we  find  that  on  May  25  he  did 
not  owe  $625  (the  face  amount  of  his  debt),  but  only  $625  (the  face  less  $3.0599  discount).  If  there 
were  no  credits  to  be  considered  on  May  25,  1907,  he  would  owe  $621.94  as  a cash  balance.  However, 
if  on  May  25  he  be  credited  for  $200  not  paid  until  June  12,  he  should  be  charged  interest  on  that  sum 
for  the  18  days  between  May  25  (when  he  received  credit  for  its  payment)  and  June  12  (when  such 
payment  was  actually  made) ; that  is,  he  should  be  charged  with  60  cents  interest  on  this  item.  And.  if  on 
May  25  he  receives  credit  for  the  $300  (the  payment  not  made  until  August  30),  he  should  be  charged 
interest  on  this  item  for  the  97  days  between  May  25  (when  he  received  credit  for  its  payment)  and 
August  30  (when  it  was  actually  paid)  ; that  is,  he  should  be  charged  with  $4.85  interest.  Thus,  we 
find  that  ou  May  25  he  should  have  received  credit  for  the  sum  of  his  payments  $500  less  the  interest. 
$5.45  charged  against  him,  or  for  $494.55,  as  a cash  balance  ; that  is,  May  25  he  owes  $125  and  was 
charged  with  the  interest  balance  of  the  difference  between  $5.45  and  $3.0599,  or  $2.3901.  In  other 
words,  on  May  25,  1907,  he  not  only  owed  $125  balance  of  items,  but  also  $2.39  balance  of  interest  ; 
that  is,  he  had  been  owing  $125  for  a length  of  time,  sufficient  to  enable  that  sum  to  accumulate  $2.3901 
interest.  Divide  this  interest  by  the  interest  on  the  balance  for  one  day  and  find  that  on  May  25,  1907, 
Butler  had  been  owing  $125  for  115  days. 

Counting  back  115  days  from  May  25,  1907,  we  find,  as  before,  the  balance  has  been  due  by 
equation  since  January  30. 


EQUATION  OF  ACCOUNTS 


269 


871.  Rule  .—Assume  the  earliest  due  date  in  the  account  as  a focal  date.  Find 
the  interest  ( discount ) on  each  item  on  the  debit  side  (to  four  places),  supposing  each  to 
be  paid  on  this  assumed  date  Find  the  interest  on  each  item  of  the  credit  side,  sup- 
posing each  to  be  due  on  this  assumed  date.  Find  the  difference  between  the  sum  of  the 
interests  ( discounts ) and  the  sum  of  the  interests.  Divide  this  difference  by  the  interest 
on  the  balance  of  the  account  for  one  day.  The  quotient  is  the  number  of  days  from 
this  assumed  date  to  the  date  on  which  the  balance  of  the  account  is  due.  If  the  differ- 
ence between  the  sums  of  money  and  the  difference  between  the  sums  of  interest  are  on 
the  same  side,  count  forward  ; if  on  opposite  sides,  count  backward  from  focal  date. 


Note. — The  foregoing  operations  give  only  assurance,  not  proof,  of  the  correctness  of  the  result. 
To  ascertain  this  find  the  sum  of  the  interest  on  the  debit  items  from  their  respective  dates  hack  to  the 
equated  date  ; also,  the  sum  of  the  interest  of  the  credit  items  from  their  respective  dates  back  to  the 
equated  date.  If  these  sums  are  the  same,  or  less  than  one-half  of  the  interest  (.0104)  of  the  balance 
$125  for  one  day,  we  have  verified  the  conclusions  in  Art.  868  and  Art.  870  ; and  the  balance  of  the 
account  is  due  January  30,  1907.  Study  the  following  operation. 


Proof  by  Interest  Method 
Equated  date  Jan.  30,  1907. 


1907 

DUE  DATE  AMT. 

D. 

INTEREST 

1907 

DUE  DATE 

AMT.  D. 

INTEREST 

May  15 

May  ! 25  285 

115 

5.4625 

June  12 

June  12 

200  133 

4.4333 

June  18 

July  18  340 

169 

9.5766 

July  1 

Aug.  30 

300  212 

10.6000 

625 

15.0391 

500 

15.0333 

500 

15.0333 

125 

.0058 

Cash  Balance 

872.  The  Cash  Balance  of  an  account  current  is  the  amount  due  at  a 
specified  date. 

873.  An  Account  Current  is  a written  statement  of  the  gross  debits  and 
credits  resulting  from  business  transactions,  within  a certain  period,  between  two 
houses.  It  may  be  rendered  monthly,  quarterly,  semiannually,  or  annually. 
Interest  is  usually,  though  not  invariably,  charged ; that  being  regulated  by  a 
previous  understanding,  or  by  the  custom  of  each  particular  business.  Since 
the  interest  that  is  added  also  earns  interest,  the  oftener  an  account  is  adjusted 
and  interest  added,  the  greater  the  balance  or  amount  due.  Some  merchants, 
however,  who  render  accounts  current  oftener  than  once  a year,  do  not  carry  the 
interest  balance  to  the  main  column  of  the  account  until  the  end  of  the  year. 

A Cash  Balance  may  be  ascertained  in  either  of  two  ways  : (1)  the  inter- 
est method;  (2)  the  equation  method. 


270 


EQUATION  OF  ACCOUNTS 


874.  To  find  the  cash  balance  of  an  account. 

Example. — Find  cash  balance  June  20,  1908. 


Dr.  John  R.  Cartman  & Co.  in  % Current  with  E.  H.  Rice  O. 


1908 

1908 

Mar. 

9 

Mdse.,  net 

600 

00 

Apr. 

1 

Cash 

550  00 

U 

25 

“ 60  da. 

796 

49 

May 

2 

Note,  10  da. 

695  00 

Apr. 

14 

“ 2 mo. 

1472 

74 

Explanation — In  this  operation  after  maturing  the  debit  and  credit  items  we  first  find  the 
number  of  days  that  each  debit  and  each  credit  item  has  been  due  prior  to  June  20,  1908,  and  compute 
the  interest  on  the  same  for  this  time  at  6%.  We  then  take  the  difference  between  the  interests  of  the 
debit  and  the  credit  items  ; and,  as  the  excess  of  balance  is  against  Cartman  & Co.,  we  charge  it  to 
them.  To  find  the  cash  balance,  take  the  difference  between  the  debit  and  the  credit  amounts  of  the 
account. 


875.  Rule. — (1)  Find  the  interest  on  each  item  on  the  debit  side  from  its  due  date 
to  the  date  of  settlement ; then  find  the  interest  on  each  item  on  the  credit  side  from  its 
date  to  the  date  of  settlement . Add  the  sum  of  these  interest  items  to  their  respect) re 
sides,  and  the  balance  will  be  the  amount  due  on  the  date  of  settlement. 

(2)  Find  the  interest  on  the  balance  of  the  account  from  the  equated  date  to  the 
date  of  settlement;  if  the  date  of  settlement  is  later  than  the  equated  date,  add  this 
interest  to  the  balance  ; if  before,  subtract  it. 

Note. — The  first  method  is  exact ; the  second  sufficiently  accurate  for  many  practical  purposes. 


EQUATION  OF  ACCOUNTS 


271 


876.  To  equate  an  account  current,  proceed  as  in  a former  explana- 
tion of  this  subject. 

Remark. — The  following  example  is  to  show  how  to  adjust  an  Account  Current  in  cases  in 
which  one  or  more  items  fall  due  after  the  date  of  settlement.  In  business  practise  it  is  the  custom  of 
bookkeepers  to  place  the  interest  of  items  which  fall  due  after  the  date  of  taking  the  cash  balance  in 
the  opposite  interest  column  ; such  entries  being  known  as  contra  entries. 

Example. — Find  cash  balance  February  1,  1907. 


Dr.  Bernard  F.  Lewis  Or. 


1906 

Nov. 

4 

Mdse.,  net 

762 

85 

1906 

Nov. 

11 

Cash 

700 

00 

<< 

18 

“ 30  da. 

1295 

62 

“ 

26 

Note,  10  da. 

1250 

00 

Dec. 

15 

“ 2 mo. 

1624 

75 

Dec. 

19 

“ 60  da. 

695 

00 

1907 

Jan. 

14 

“ 4 mo. 

269 

55 

1907 

Jan. 

19 

“ 2 mo. 

360 

00 

“ 

28 

“ net 

974 

86 

Example. — Vim  Brothers  in  account  current  with  Enterprise  Mill  Co.,  July  1,  1907 


272 


EQUATION  OF  ACCOUNTS 


o 


JZ  CD 


05  ~ c- 


s 


05  C 


3 

^5 


o 

M 

o 

v 

Cm 


1m 

CQ 

e-i 

iz 

K 

« 

P 

o 

H 

5S 

P 

5 

o 

C 

fs 

«! 


o 


CQ 


< 

o 


$10.88  ba.1  of  i ut.  1 551290 — 90000=05290  balance  of  products. 


EQUATION  OF  ACCOUNTS 


273 


WRITTEN  PROBLEMS 

Remark. — When  first  presenting  this  subject  the  teacher  may  use  any  method  he  prefers,  but  in 
review  it  will  be  well  to  familiarize  the  student  with  the  other  methods.  Each  method  has  some  point 
of  merit.  The  teacher  may  have  the  student  find  the  equated  date  by  one  method  and  verify  the  result 
by  another  method  ; or  the  equated  date  may  be  found  by  the  particular  method  under  consideration 
and  the  result  proved.  The  fact  should  be  emphasized  that  the  student  may  assure  himself  by  verifica- 
tion or  proof  that  he  has  obtained  the  correct  date. 

877.  When  are  the  following  accounts  due  by  equation? 


1.  .John  Hardegg 


To  William  Price,  Dr. 


1908 

March 

2 

To  Mdse. 

375 

00 

U 

18 

U 

1565 

00 

it 

31 

u cc 

742 

00 

|. 

James  Martin 

To  Philip  Shellening,  Dr. 

1908 

Feb. 

6 

To  Mdse. 

496 

72 

23 

u a 

962 

85 

Mar. 

10 

a u 

1742 

10 

3. 

Henry  Hall 

To  James  R.  Mick  ley,  Dr. 

1908 

June 

8 

Mdse.,  60  da. 

492 

67 

a 

16 

“ 10  11 

1275 

43 

a 

30 

“ 15  “ 

674 

92 

Packer  IJ  Goodfiall 

To  McFetridge  & Hall,  Dr. 

1908 

| 

July 

16 

Mdse.,  2 mo. 

1129 

63 

(( 

28 

u -j^  u 

976 

42 

Aug. 

4 

“ 3 “ 

497 

12 

u 

18 

“ 60  da. 

794 

13 

5.  First  find  equated  date,  then  find  amount  due  July  6,  1908. 


Goodyear  & Co. 

To  William  H.  Heaney,  Dr. 


1908 

April 

ki 

8 

Mdse.,  120  da. 

796 

45 

13 

“ 3 mo. 

1726 

50 

a 

28 

“ 60  da. 

475 

22 

May 

12 

“ 30  “ 

762 

95 

June 

6 

“ 1 mo. 

742 

94 

274 


EQUATION  OF  ACCOUNTS 


6.  Find  when  the  balance  of  the  following  account  is  due  by  equation,  and 
the  amount  due  June  J , 1909. 

William  Sanford 


To  Freihofer  & Bro.,  Dr. 


1908 

Sept. 

2 

To  Mdse.,  1 mo. 

120 

00 

Oct. 

9 

“ “ 60  da. 

340 

75 

Nov. 

15 

“ “ 3 mo. 

278 

93 

Dec. 

18 

“ “ 30  da. 

200 

00 

1909 

Feb 

28 

“ “ 2 mo. 

362 

84 

7.  Find  when  the  balance  of  the  following  account  is  due  by  equation. 


Dr.  George  Scarlet  Cr. 


1908 

Sept. 

12 

Mdse. 

450 

o 

o 

1908 

Oct. 

1 Cash 

Nov. 

10 

(( 

525 

00 

Dec. 

1 

450  00 
500  00 


8.  Find  when  the  balance  of  the  following  account  is  due  by  equation. 


Dr.  I.  Merchant  Cr. 


1908 

Oct. 

1 

Cash 

450! 

00 

I 1908 
j Sept. 

12 

Mdse. 

450  00 

Dec. 

1 

U 

500 

|00; 

| Nov. 

10 

CC 

525  00 

9.  Find  when  the  balance  of  the  following  account  is  due  by  equation. 


Dr. 


David  Deal  & Bro. 


Or. 


1908 

Apr. 

June 


1908 

1 

Cash 

1400 

00, 

Jan. 

3 

1 

u 

1200 

00 

Feb. 

2 

- 

Mar. 

4 

Apr. 

5 

Mdse. 

(( 


U 


726  00 
57 4 1 '<> 
267  00 
472  00 


10.  Find  when  due  by  equation,  and  the  amount  that  would  settle  the 
balance  on  the  latest  date  of  account. 


Dr.  Beyer  Brothers  Cr. 


1908 

Jan. 

3 

Mdse.,  30  da. 

67925 

j 1908 

Feb. 

3 

Cash 

o 

o 

so 

Feb. 

18 

“ 30  da. 

79420 

Mar. 

IS 

(( 

750 

EQUATION  OF  ACCOUNTS 


275 


11.  When  is  the  balance  of  the  following  account  due  by  equation  ? 

Dr.  Frank  S.  M islet  Cr. 


1908 

Mar. 

3 

Mdse.,  30  da. 

796 

14 

1908 

Apr. 

1 

Note,  60  da. 

750 

00 

it 

17 

“ 2 mo. 

1722 

11 

tt 

28 

“ 30  “ 

850 

00 

tt 

31 

“ 60  da. 

1946 

75 

May 

1 

Acceptance,  2 mo. 

1500 

00 

May 

19 

“ 30  da. 

472 

11 

June 

26 

Cash 

390 

00 

IS.  In  the  following,  find  the  amount  due  April  26,  1908. 

Dr.  J.  L.  Lippincott  Or. 


1908 

Jan. 

7 

Mdse.,  net 

826 

54 

1908 

Jan. 

15 

Cash 

675 

00 

tt 

15 

“ 30  da. 

1692 

74| 

tt 

28 

Note,  3 mo. 

1000 

00 

Feb. 

12 

“ 2 mo. 

1749 

89 

Mar. 

18 

Draft,  30  da. 

1200 

00 

“ 

25 

“ 90  da. 

1845 

29 

Apr. 

22 

Note,  4 mo. 

2000  00 

Mar. 

8 

“ 1 mo. 

749 

86| 

• 

13.  In  the  following,  find  the  equated  date,  prove  it,  and  then  find  the  cash 
balance  Sept.  15,  1908. 

Dr.  John  N.  Chapman  Or. 


1908 

1908 

May 

6 

Casli 

1000 

00 

May 

6 

Mdse.,  net 

1620 

70 

it 

20 

Note,  2 mo. 

1620 

00 

June 

12 

“ 60  da. 

922  40 

June 

15 

Acceptance,  60  da. 

2240 

00 

Aug. 

19 

“ 2 mo. 

1452  49 

J uly 

22 

Cash 

1490 

00 

“ 

27 1 “ 4 mo. 

2975 14 

U- 

Find  cash  balance  June  1,  1908  ; find  also  the  equated  date. 

Dr. 

John  R. 

Cartman 

& 

Co. 

Cr. 

1908 

j:  1908 

! 

Mar. 

9 

Mdse.,  net 

600 

00 

Apr. 

1 

Cash 

550  00 

tc 

25 

“ 60  da. 

796  49 

II  May 

2 

Note,  10  da. 

695  00 

Apr. 

14 

“ 2 mo. 

1472  74 

15. 

Find  cash  balance  April  15,  1909. 

Dr. 

James 

J.  King  & Co. 

Or. 

1908 

1908 

Aug. 

12 

Mdse.,  30  da. 

796 

41 

Sept. 

23 

Cash 

1000 

00 

Sept. 

8 

1 mo. 

949 

56 

Nov. 

17 

Note,  2 mo. 

1200 

00 

Nov. 

17 

“ 60  da. 

1264 

75 

Dec. 

22 

Acceptance,  90  da. 

750 

00 

Dec. 

15 

“ 2 mo. 

694 

86 

1909 

1909 

Mar. 

6 

Note,  30  da. 

1420 

00 

Feb. 

10 

“ 90  da. 

1543 

75 

270 


EQUATION  OF  ACCOUNTS 


16.  What  is  the  cash  balance  due  Jan.  27,  1909;  when  due  by  equation? 


Dr.  John  Mills  & Co.  Cr. 


1908 

1908 

Sept. 

5 

To  Mdse. 

60  da. 

758 

90 

Oct, 

4 

By  Cash 

1000  00 

CC 

18 

CC 

u 

30  “ 

387 

40 

Nov. 

7 

cc  cc 

520 

00 

Oct. 

7 

CC 

cc 

4 mo. 

500 

00 

cc 

30 

“ Draft,  30  da. 

100 

00 

cc 

20 

cc 

cc 

10  da. 

400 

50 

Dec. 

18 

Cash 

650 

75 

Nov. 

6 

cc 

cc 

3 mo. 

515 

70 

1909 

u 

29 

Ci 

cc 

net 

400 

00 

•Jan. 

15 

By  acceptance,  60  da. 

58400 

Dec. 

8 

cc 

a 

90  da. 

274 

60 

Feb. 

9 

“ Cash 

478 

90 

CC 

27 

u 

cc 

2 mo. 

309 

75 

Mar. 

10 

CC  CC 

35040 

1909 

Jan. 

14 

cc 

cc 

1 mo. 

357 

80 

May 

2 

cc  cc 

150 

00 

Feb. 

5 

cc 

cc 

30  da. 

200 

00 

17.  When  are  the  proceeds  of  the  following  account  sales  due  by  equation? 


Philadelphia,  Pa.,  Apr.  7,  190S. 


Account  sales  of  flour  sold  for  account  of  J.  S.  Harlington  & Co. 


1908 

Feb. 

11 

420  bbl.  at  §4.10 

CC 

25 

245  “ “•  4.62 

Mar. 

18 

296  “ “ 4.48 

CC 

31 

369  “ 4.32 

Apr. 

7 

162  “ “ 4.25 

Charges 

Feb. 

14 

F reight 

89 

15 

CC 

18 

Cartage 

16 

75 

Mar. 

20 

Remittance 

200000 

Apr. 

7 

Storage 

100  00 

CC 

7 

Cooperage 

12 

00 

Commission  4% 

Note. — Date  the  commission  on  the  last  date  of  sales,  and  equate  the  account  in  the  ordinary 
way,  taking  the  charges  as  debits  and  the  sales  as  credits. 


EQUATION  OF  ACCOUNTS 


277 


18.  Account  sales  of  shoes 

Sold  for  account  of  Vm.  B.  Munich  & Co. 


1908 

July 

7 

325  pairs  at  $2.10,  cash 

(C 

22 

456  “ “ 2.55,  3 mo. 

Aug. 

19 

267  “ “ 1.75,  60  da. 

U 

31 

412  “ “ 2.18,  cash 

Sept. 

24 

564  “ “ 2.95,  90  da. 

Charges 

July 

9 

Freight 

102 

00 

Aug. 

10 

Dravage 

5 

60 

U 

26 

Remittance 

1000 

00 

Sept. 

24 

Storage 

110 

00 

Commission  5% 

19.  Account  sales  of  broadcloth 


Sold  for  account  of  John  Detts’  Sons 


1908 

May 

5 

375  yards 

at 

$3.25,  cash 

“ 

20 

642 

(£ 

u 

3.92,  60  da. 

June 

17 

895 

U 

u 

4.15,  3 mo. 

(( 

30 

268 

U 

u 

3.68,  net. 

July 

15 

495 

u 

u 

4.10,  30  da. 

Aug. 

14 

722 

u 

a 

4.05,  60  da. 

Charges 

May 

5 

Inspection 

9 

00 

May 

19 

Freight 

59 

20 

June 

2 

Cartage 

10 

50 

July 

10 

Remittance 

2500 

00 

Aug. 

14 

Storage 

90 

25 

Commission  3% 

Guaranty  2% 

Net  proceeds  due 

30.  Account  sales  of  apples 

Sold  for  account  of  Joshua  Half.y 


1908 

Oct. 

8 

228  bbl.  at  $2.25,  cash 

U 

22 

245  “ “ 2.30,  10  da. 

Nov. 

2 

280  “ “ 2.40,  30  da. 

U 

18 

175  “ “ 2.28,  10  da. 

Charges 

Oct. 

5 

Freight  and  drayage 

92 

50 

Nov. 

18 

Storage 

18 

56 

Oct. 

5 

Advertising  and  Insurance 

25 

00 

Commission  2J%  Guaranty  2% 

Net  proceeds  due 

| 

ACCOUNTS  BEARING  INTEREST 


ACCOUNTS  BEARING  INTEREST 
Daily  Balances 

The  Frugality  Sayings  Company 

In  account  with  John  Wise. 


1908 

1908 

July 

1 

To  Balance 

155 

00 

July 

1 

By  Check 

33 

33 

U 

9 

“ Cash 

50 

00 

a 

15 

a a 

15 

00 

a 

23 

a a 

50 

00 

Aug. 

1 

a a 

33 

33 

Aug. 

6 

£(  u 

50 

00 

a 

15 

a a 

12 

00 

U 

20 

a a 

50 

00 

Sept. 

1 

a a 

33 

33 

Sept. 

10 

a u 

50 

00 

U 

15 

a a 

10 

00 

U 

24 

(i  a 

50 

00 

Oct. 

1 

“ Balance 

319 

70 

Oct. 

1 

“ lilt. 

1 

69 

456 

69 

456 

69 

1908 

Oct. 

1 

To  Balance 

319 

70 

Operation 

Balances 

Days 

$121.67  from 

■July  1 

to  J 

ul\ 

9 

8 

$968 

171.67  “ 

“ 9 

U 

(i 

15 

6 

1026 

156  67  “ 

“ 15 

a 

C 

23 

8 

1248 

206.67  “ 

“ 23 

“ Allg- 

1 

9 

1854 

1 73  34  “ 

Aug.  1 

a 

6 

5 

S65 

223.34  “ 

“ 6 

a 

u 

15 

9 

2007 

211.34  “ 

“ 15 

a 

u 

20 

5 

1055 

261.34  “ 

“ 20 

“ Sept 

1 

12 

3132 

228.01  “ 

Sept.  1 

U 

u 

10 

9 

2052 

278.01  “ 

“ 10 

u 

u 

15 

5 

1390 

268.01 

15 

o 

a 

24 

9 

2412 

318.01  “ 

24 

“ Oct 

1 

7 

2226 

$20235 

Total  daily  balance,  $20235  for  1 day 
Interest  at  3%  $1.69 


ACCOUNTS  BEARING  INTEREST 


279 


Form  with  Balance  Columns 


Penn  Trust  Company 

In  account  with  Smith  & Jones. 


Date.  Dr.  Cr.  Daily  Average.  Days. 


1908 

Jan. 

1 

8000 

00 

8000 

00 

14 

U 

15 

2000 

00 

10000 

00 

16 

a 

31 

5000 

00 

5000 

00 

15 

Feb. 

15 

6000 

00 

11000 

00 

13 

U 

28 

4500 

00 

6500 

00 

15 

JMar. 

15 

4 00 

00 

11000 

00 

16 

31 

5000 

00 

6000 

00 

1 

20500 

00 

14500 

00 

Apr. 

1 

42 

75 

Bal. 

601$ 

75 

20542 

75 

20542 

75 

Apr. 

T 

6042 

75 

Interest  at  2%  on  1769500  for  1 day=$42.75. 


Total  Balance. 


112000 

00 

160000 

00 

75000 

00 

143000 

00 

97500 

00 

176000 

00 

6000 

00 

769500 

00 

878.  It  is  to  be  noticed  that  the  interest  rates  differ  according  to  the  cus- 
toms of  various  lines  of  business.  Trust  companies  commonly  pay  2%  on  daily 
balances  on  active  accounts ; on  those  that  require  notice  to  withdraw  money,  it 
is  common  to  allow  3%  on  daily  balances. 

In  stock  brokers’  accounts  with  their  customers,  and  on  other  running 
accounts  bearing  interest,  the  legal  rate  is  charged,  unless  there  is  a special 
agreement  upon  some  other  rate.  It  sometimes  happens  that  one  side  of  an 
account  bears  one  rate  and  the  other  side  another  rate ; as,  where  a broker 
■charges  6%  on  debit  balances  and  allows  3%  on  credit  balances. 

If  an  item  is  not  yet  due  at  the  day  of  settlement,  it  is  usually  adjusted  by 
a contra  entry  ; that  is,  by  adding  the  interest  to  the  other  side  of  the  account. 


WRITTEN  PROBLEMS 

879.  1.  Wh  at  is  the  net  amount  due  on  the  following  account  July  1, 1908  ? 


Dr.  I.  M.  Porter  in  account  current  with  Vici  Brothers.  Cr. 


1907 

1907 

• 

July 

1 

Balance 

1200  00  ■ Nov. 

1 

Mdse.,  3 mo. 

600 

00 

Sept. 

12 

Draft 

900 

00  ;j  1908 

1908 

Mar. 

14 

“ 60  da. 

600 

00 

-J  an . 

3 

U 

900 

00!  Apr. 

25 

“ 30  da. 

850 

00 

May 

16 

u 

400 

00|j  June 

4 

“ net 

500 

00 

280 


ACCOUNTS  BEARING  INTEREST 


2.  What  is  the  balance  of  the  following  account  May  1, 1908,  interest  at  6%  ? 
Dr.  Hill  & Son  in  account  with  Grove  Land  Company.  Or. 


1908 

Jan. 

15 

Draft 

950 

00 

1908 

Jan. 

1 

Balance 

3500 

Feb. 

1 

a 

860 

00 

U 

22 

Sales  at  3 mo. 

1200 

Mar. 

1 

u 

1800 

00 

Feb. 

14 

“ 3 “ 

700 

“ 

24 

u 

960 

00 

Mar. 

2 

“ 3 “ 

2400 

3.  The  City  Trust  Company  allows  2%  interest  on  daily  balances  of  §100  or 
more.  What  is  the  balance  to  the  credit  of  Charles  Careful  on  December  31, 
1908,  after  crediting  him  with  the  interest  due? 

Deposits:  March  5,  1908,  $50;  April  1,  1908,  $40;  April  15,  1908,  $60; 
April  30,  1908,  $50  ; May  14, 1908,  $50  ; June  1,  1908,  $35  ; June  15,  1908,  $50  : 
June  25,  1908,  $40;  June  27,  1908,  $35;  July  16,  1908,  $52.92;  July  9,  1908, 
$62.50;  July  30,  1908,  $50;  August  3,  1908,  $38;  August  16,  1908,  §50;  August 
31,  1908,  $50;  September  15,  1908,  $40;  September  30,  1908,  $40;  October  15, 
190S,  $50  ; October  31,  1908,  $50  ; November  25,  1908,  $50. 

Checks:  March  19,  1908,  $12;  April  9,  1908,  $10;  April  15,  1908,  $25  ; 
April  21,  1908,  $20  ; April  23,  1908,  $12  ; May  2,  1908,  $10  ; May  7,  1908,  §11  : 
May  14,  1908,  $25 ; May  20, 1908,  $10  ; May  25,  1908,  $10 ; June  10,  1908,  $10  ; 
June  15,  1908,  $25  ; June  20,  1908,  $10  ; June  24, 1908,  $20  ; July  1,  1908,  $100  ; 
July  5,  1908,  $33.33  ; July  20,  1908,  $25;  July  27,  1908,  $7  ; August  1,  1908, 
$33.33;  August  5,  1908,  $20;  August  10,  1908,  $11.50  ; August  12,  1908,  $10; 
August  13,1908,  $10;  August  17,  1908,  $15;  August  24,  1908,  $15;  September 
1,  1908,  $33.33;  September  1,  1908,  $10  ; September  1,  1908,  $19.55  ; September 
15,  1908,  $10  ; September  21,  1908,  $15;  September  30,  1908,  $3.25;  October  3, 
1908,  $33  33;  October  7,  1908,  $16.80;  October  8,  1908,  $10;  October  14,  1908, 
$10;  October  21,1908,  $12;  October  26,  1908,  $20;  October  28,  1908,  §8; 
November  2,  1908,  $5;  November  3,  1908,  $33.33;  November  6,  1908,  $10: 
November  9,  1908,  $4.70;  November  12,  1908,  $6. 

880.  To  calculate  the  interest  on  a saving-fund  account. 

WRITTEN  PROBLEMS 

881.  In  the  Western  Saving  Fund  Society  of  Philadelphia  interest  is  allowed 
at  the  rate  of  three  and  one-half  per  cent,  per  annum  on  accounts  of  five  dollars 
and  over.  No  interest  on  fractional  parts  of  a dollar.  Interest  is  calculated  by 
calendar  months,  and  is  allowed  on  deposits  made  on  or  before  the  fifth  of  the 
month  ; in  the  case  of  withdrawals,  interest  is  allowed  for  one-half  month  on 
money  withdrawn  on  or  after  the  sixteenth  of  the  month.  At  the  end  of  the 
calendar  year  the  interest  is  reckoned,  and  is  added  to  the  principal. 

1.  William  Trask  deposited  in  the  Western  Saving  Fund  Society  of  Phila- 
delphia, as  follows:  January  2,  1907,  $20;  February  16,  1907,  $24;  March  14. 
1907,  $18;  April  26,  1907,  $60;  May  31,  1907,  $12  ; June  3,  1907,  $15;  July  19, 
1907,  $36;  August  24,  1907,  $72  ; November  14,  1907,  $24;  December  6,  1907, 
$50. 


ACCOUNTS  BEARING  INTEREST 


281 


On  March  18,  1907,  he  withdrew  $16;  April  10,  1907,  $15;  September  9, 
1907,  $25;  October  3,  1907,  $15.  What  was  the  balance  of  his  account  January 
1,  1908? 

2.  Mary  Jones  deposits  in  the  Western  Saving  Fund  Society  as  follows: 
February  3,  1904,  $20;  May  6,  1904,  $32;  July  7,  1904,  $18;  October  22,  1904, 
$22;  December  15,  1904,  $25;  February  10,  1905,  $15;  April  20,  1905,  $17; 
June  30,  1905,  $24:  August  17,  1905,  $12  ; September  26,  1905,  $20  ; November 
3,  1905,  $15;  December  20,  1905,  $21;  February  9,  1906,  $16;  April  14,  1906, 
$25;  June  12,  1906,  $18  ; August  22,  1906,  $20  ; October  12,  1906,  $16;  Decem- 
ber 5,  1906,  $25;  January  26,  1907,  $12;  March  12,  1907,  $26;  May  18,  1907, 
$17;  July  16,  1907,  $30;  September  18,  1907,  $12;  November  7,  1907,  $21  ; 
December  20,  1907,  $14  ; January  24,  1908,  $10 ; March  30,  1908,  $20.  What  is 
the  balance  due  her  on  July  1,  1908? 

3.  On  January  1,  1907,  William  Smith’s  balance  in  a savings  bank,  in 

which  interest  at  3%  per  annum  is  allowed  on  the  smallest  monthly  balance  and 
compounded  quarterly,  was  $180.  During  the  year  he  made  the  following 
deposits:  January  18,  $24;  February  6,  $16;  March  4,  $28;  March  6,  $14; 

March  30,  $15 ; April  16,  $13  ; May  1,  $60 ; May  8,  $24 ; May  16,  $25 ; June  3, 
$12  ; June  20,  $18  ; July  10,  $18  ; August  1,  $14  ; August  20,  $13  ; September  3, 
$16;  October  24,  $28;  November  5,  $22;  December  3,  $30.  He  withdrew 
January  21,  $25  ; February  28,  $30  ; April  20,  $60;  August  20,  $25  ; September 
16,  $12 ; October  31,  $15;  December  24,  $12.  What  was  the  balance  of  the 
above  account  January  1,  1908  ? 

J.  Mr.  W.  T.  Zaner  made  the  following  deposits  and  withdrawals  at  a sav- 
ings bank,  the  by-laws  of  which  provide  for  an  allowance  of  interest  at  three 
per  cent,  per  annum  on  the  smallest  balance  for  interest  terms  of  three  months 
ending,  respectively,  January  1,  April  1,  July  1,  and  October  1,  interest  to  be 
compounded  semiannually.  (That  is,  a deposit  begins  to  draw  interest  from  the 
first  day  of  each  quarter,  and  only  the  balance  left  the  entire  quarter  draws 
interest  in  the  event  of  withdrawals.)  He  deposited  January  1.  1907,  $600  ; 
February  2,  1907,  $145  ; February  15,  1907,  $180;  March  16,  1907,  $160;  March 
28,  1907*  $240;  April  20,  1907,  $160;  May  1,  1907,  $120;  June  21,  1907,  $180; 
July  16,  1907,  $125;  August  26,  1907,  $280;  September  2,  1907,  $460;  October 
4, 1907,  $240;  October  18,  1907,  $160;  November  12,  1907,  $180.  He  withdrew 
January  22,  1907,  $100  ; February  28,  1907,  $200  ; March  25,  1907,  $200 ; May 
20,  1907,  $150;  July  1,  1907,  $50;  October  23,  1907,  $20;  December  24,  1907, 
$200.  What  was  due  Mr.  Zaner  January  1,  1908  ? 


BANKRUPTCY 

882.  A bankrupt  or  insolvent  is  a debtor  whose  property  is  taken  to  be 
divided  among  his  creditors  by  a court  under  the  operation  of  an  insolvent  law. 

883.  An  insolvent  law  provides  that  when  a person  has  been  judicially 
ascertained  to  be  insolvent,  and  has  surrendered  his  property  for  distribution 
among  his  creditors  under  a decree  of  court,  he  is  to  be  discharged  from  his 
indebtedness. 

884.  An  assignee  in  bankruptcy,  or  an  assignee  in  insolvency,  is  a 

person  to  whom  the  property  of  a bankrupt  or  insolvent  is  transferred  in  order 
that  he  may  dispose  of  the  same  for  the  benefit  of  the  creditors. 

885.  A dividend  is  a sum  arising  out  of  the  bankrupt’s  available  assets, 
which  the  assignee  divides  among  the  creditors  pro  rata. 

886.  To  find  each  creditor’s  share. 

Example. — The  statement  of  a bankrupt’s  affairs  was  as  follows: 

Assess:  Cash,  $2870.13;  Bills  Receivable,  $3750,  Merchandise,  $8439.70 ; 
Real  Estate,  $4200  ; Stocks  and  Bonds,  $3430.75;  Personal  Accounts,  $7833.48. 

Liabilities:  Due  A,  $9238.17;  clue  B,  $12375;  due  C,  $20453.38;  due  D, 
$17216.85;  due  E,  $4114.29.  The  expenses  of  the  assignment  were  $917.34. 
What  was  the  rate  of  dividend,  and  how  much  should  each  creditor  receive? 


$2  270. /3 

r 37s  o. 

3 337.76? 

2 20  O, 

3 23  0.7  s 
7 23  3.  32 
O >2  2 3 • O 6 

2/ 7.3  3 

6 o 6.7  2 


$723227 

2237s. 

2 0 2S3.  32 
/7 2/ 7.  7S 
3/63.2  7 


3 3 y y.3  7 


$276,0 7. 72  -2 $63377.67  = .267^ 

$ 7 23 7. 7 7 7,267  = $23/ 2.23 
/237S.  7.267=  S777 ./ 2 73s 

2 0333.37 X .267  = 7 SS/.  73  672 

/ 72/  6.237.267  = S 020. 27 As 
2/ / 2.277  ,267  = / 22/. 37 

$27606.72 

UT‘ , 


/ 63377.67  : $7237/7 

63377.67  : /237S.  OO 
63  3 77.67  : 2 0 333.37 

63377.67  : / 72/ 6.73 

63377.67  : 27/2.27 


: : $2760  6.72  : 3/^^ 
::  27606.72:732°,, 

::  2760  6.72  : 62  v 
::  27606  72  : 7h  „ 

::  2760  6 72  : & „ 


= 7/23/3.23 

J 377722 
= 7SS/.73 
_ 7030.27 
= Z72/.37 


Note — Usually  it  is  necessary  to  carry  the  decimal  to  eight  places. 

282 


BANKRUPTCY 


283 


887.  Rule. — Deduct  expenses  of  the  assignment  from  the  total  assets,  and  divide 
by  the  total  liabilities.  The  quotient  will  be  the  rate  of  dividend.  Then  multiply  the 
amount  of  each  creditor's  claim  by  the  rate  of  dividend.  Or,  by  proportion, 

Total  liabilities:  Each  liability : : Net  assets  : Each  creditor's  share  of  assets. 

WRITTEN  PROBLEMS 

888.  l.  If  a firm  fails  with  liabilities  amounting  to  $60000  and  assets 
$42000,  how  much  should  each  creditor  receive  on  the  dollar — expenses  of  assign- 
ment being  $1412?  What  sum  should  a creditor  receive  whose  claim  amounted 
to  $6489.78? 

2.  A bankrupt’s  assets  are  as  follows  : Cash,  $1263.18  ; Merchandise,  $6492.72  ; 
Bills  Receivable,  $3820;  Personal  Accounts,  $9868.47.  His  liabilities  are  : Due 
to  Jones,  $20316.24;  Brown,  $11624.25;  Smith,  $7422.96;  Clark,  $3127.89. 
Expenses  of  settlement,  $812.70.  How  much  does  each  creditor  receive? 

3.  After  converting  all  the  available  assets  into  cash  and  paying  all 
expenses,  an  assignee  has  in  his  hands  $73462.37  to  distribute  among  creditors 
whose  claims  amount  to  $104392.64.  How  much  does  A receive,  whose  claim 
is  $21622.78  ? 

J.  An  insolvent  merchant  owes  A,  $3122.75  ; B,  $2646.38  ; C,  $9421.67 ; 
D,  $248.90 ; E,  $647.28 ; F,  $427.32.  His  assignee  has  realized  $9862.48  from  the 
assets,  and  has  $364.77  expenses  to  pay.  How  much  will  each  of  the  creditors 
receive  ? 


PARTNERSHIP  SETTLEMENTS 

889.  A partnership  is  an  association  of  individuals  formed  by  an  agreement 
made  between  them  to  combine  their  money,  labor  and  skill  in  some  business 
enterprise,  the  profits  and  losses  of  which  are  divided  in  certain  proportions. 
This  association  is  generally  called  a firm  or  house. 

890.  The  capital  of  a partnership  consists  of  the  money,  real  estate,  or  any 
other  property  invested. 

891.  The  resources  of  a firm  consist  of  the  money  and  other  property  it 
owns,  and  the  debts  due  the  firm.  In  theory,  the  firm  is  distinct  from  the  indi- 
viduals composing  it ; hence,  a partner  may  owe  the  firm,  and  his  obligation  to 
the  firm  is  a resource  of  the  firm,  as  in  the  case  of  withdrawals. 

892.  The  liabilities  of  a firm  are  the  debts  it  owes.  For  accounting  pur- 
poses the  firm  may  be  considered  as  owing  the  individuals  composing  the  firm 
for  their  investment;  hence,  the  capital  of  the  business  is  a quasi  liability  of  the 
firm. 

893.  The  net  capital  of  a firm,  or  present  worth,  is  the  excess  of  the 
resources  over  the  liabilities.  While  the  excess  of  resources  over  liabilities  is 
called  either  net  capital  or  present  worth,  still,  strictly  speaking,  the  former  is 
the  appropriate  term  to  use  when  opening  books  and  the  latter  when  closing. 

894.  When  a firm’s  resources  exceed  its  liabilities,  it  is  said  to  be  solvent. 
When  the  liabilities  exceed  the  resources,  the  firm  is  said  to  be  insolvent. 


284 


PARTNERSHIP  SETTLEMENTS 


895.  The  net  insolvency  of  a firm  is  the  excess  of  liabilities  over 
resources. 

896.  The  net  gain  is  the  excess  of  gain  over  loss  for  a given  period ; or  the 
excess  of  present  worth  over  the  invested  capital. 

897.  The  net  loss  is  the  excess  of  loss  over  gain  for  a given  period  : or  the 
excess  of  invested  capital  over  present  worth. 

898.  An  adventure,  enterprise,  or  venture,  is  usually  a co-operation  for 
a.  single  business  arrangement  for  a limited  time  or  for  separated  business 
arrangements,  and  they  are  not  partnerships  in  the  strict  use  of  that  term.  They 
are  settled  in  accordance  with  the  terms  of  the  agreement  or  in  accordance  with 
the  custom  in  such  cases. 

899.  Partnerships  are  settled  in  accordance  with  the  terms  of  the  agreement ; 
this  should  preferably  be  written,  but  may  be  oral.  The  acts  of  the  partners 
may  modify,  and,  in  fact,  entirely  abrogate  the  terms  of  the  agreement.  In 
general,  where  there  is  no  distinct  agreement  as  to  the  sharing  of  gains  or  losses, 
partners  share  equally,  no  matter  what  the  respective  amounts  of  investments 
may  be.  Differences  in  investment  are  usually  adjusted  through  interest  allow- 
ances. Where  one  or  some  of  the  members  of  a firm  are  active  in  its  business, 
devoting  their  entire  time  and  energies  to  its  interests,  while  others  are  passive, 
these  differences  are  frequently  adjusted  by  allowing  the  former  salaries. 

900.  When  two  classes  of  accounts  are  furnished,  one  showing  resources 
and  liabilities,  and  the  other  gains  and  losses,  and  the  results  obtained  do  not 
agree,  the  result  of  the  class  showing  resources  and  liabilities  is  to  be  taken,  as 
the  accounts  of  this  class  can  be  verified  by  taking  inventories,  sending  out 
statements,  etc. 

Remark. — The  principles  of  accounting  should  be  carefully  applied  in  working  the  problems 
that  follow,  and  the  statement  should  be  presented  in  such  form  that  the  results  may  be  readily  seen, 
and  the  process  by  which  they  are  attained  readily  followed.  Students  frequently  fail  to  get  accurate 
results  because  of  lack  of  system  and  method  iu  arranging  the  work. 

901.  To  find  the  net  gain  or  net  loss  of  a business  when  the  invest- 
ment is  given,  and  the  present  worth. 

Example. — 1.  What  is  the  net  gain  of  a business  whose  net  capital  at  starting 
was  $4000  and  whose  present  worth  is  $4500? 

Present  worth  $4500 

Net  capital  4000 

Net  gain  $500 

Example. — 2.  What  is  the  loss  of  a business  whose  net  capital  was  $15000, 
now  showing  a present  worth  of  $13500  ? 

Net  capital  $15000 

Present  worth  13500 

Net  loss  $1500 

902.  Rule, — Find  the  difference  between  the  net  capital  and  the  present  worth. 

If  the  present  worth  is  the  greater,  the  difference  shows  a gam  ; if  the  capital  is  the 
greater,  the  result  shows  a loss. 


PARTNERSHIP  SETTLEMENTS 


285 


WRITTEN  PROBLEMS 

903.  Find  the  net  gain  or  net  loss  in  the  following: 

1.  Net  capital  $2500,  present  worth  $3750. 

S.  Net  capital  $6000,  present  worth  $5800. 

3.  Net  capital  $15500,  present  worth  $15700. 

h Net  capital  $2500,  insolvency  $1500. 

5.  Insolvency  $500,  present  worth  $1200. 

6.  Insolvency  $200,  present  insolvency  $150. 

7.  Insolvency  $1100,  present  insolvency  $1500. 

904.  To  find  the  net  gain  or  net  loss  of  a business  when  the  items  of 
gain  and  loss  are  given. 

Example. — The  total  gains  of  a business  are  $5400  and  the  total  losses  are 
$1700;  what  is  the  net  gain  ? 

Total  gains  $5400 

Total  losses  1700 

Net  gain  $3700 

905.  Rule  . — Find  the  difference  between  the  total  gains  and  total  losses ; the 
excess  of  gain  shows  the  net  gain  and  the  excess  of  loss  the  net  loss. 

WRITTEN  PROBLEMS 

906.  1.  What  is  the  net  gain  or  loss,  if  the  total  gain  is  $3500  and  the  total 
loss  $1800? 

2.  Total  gain,  $7200;  total  loss,  $3750  ? 

3.  Total  gain,  $825 ; total  loss,  $653? 

f.  Total  gain,  $12500  ; total  loss,  $14200  ? 

5.  Total  gain,  $1700;  total  loss,  $1900? 

6.  Items  of  gain  : merchandise,  $500  ; discount  and  interest,  $80 — Item  of 
loss  : expense,  $420  ? 

7.  Items  of  gain  : merchandise,  $1250;  rent,  $400;  discount  and  interest, 
$120 — Items  of  loss  : expense,  $650  ; furniture  and  fixtures,  $35  ? 

907.  To  find  the  present  worth  of  a business  when  the  net  capital  is 
given,  and  gain  or  loss. 

Example. — A merchant  invested  in  business  $12000.  Upon  closing,  his 
Loss  and  Gain  account  shows  the  following  items  : Gains,  Merchandise,  $3500 ; 
Discount  and  Interest,  $375  . Losses,  Expense,  $1500.  What  is  the  present 
worth  of  the  business  ? 

Loss  and  Gain. 


Expense 

$1500.00 

Merchandise 

$3500.00 

Net  gain 

2375.00 

Discount  & Interest 

375.00 

$3875.00 

$3875.00 

Net  capital 

$12000 

Net  gain 

2375 

Present  worth 

$14375 

286 


PARTNERSHIP  SETTLEMENTS 


908.  Rule. — Add  the  net  gain  to  the  present  capital , or  deduct  the  net  loss 
from  it. 


WRITTEN  PROBLEMS 

909.  1.  What  is  the  present  worth,  if  net  capital  was  $3500,  gain  $550  ? 

2.  Net  capital  $7500,  loss  $700? 

3.  Net  capital  $6500,  gain  $800? 

L Net  capital  $1700,  gain  $3500? 

5.  Insolvency  $300,  gain  $600? 

6.  Insolvency  $700,  loss  $300? 

7.  Insolvency  $1200,  gain  $900? 

8.  A business  was  started  with  a capital  of  $17500;  the  totals  of  the  Loss 
and  Gain  account  show  debits  $5625,  credits  $6000.  What  is  the  present  worth  ? 

9.  The  net  capital  of  a business  was  $8000.  The  Loss  and  Gain  account 
shows  debits  : Expense,  $325  ; Discount  and  Interest,  $78  ; Real  Estate,  $200 — 
and  the  following  credits:  Merchandise,  $2250;  Stocks,  $250.  What  is  the 
present  worth  of  the  business? 

10.  A invested  $6000  and  B $3000  in  a partnership.  At  the  end  of  a year 
the  items  of  loss  were  : Expense,  $1200  ; Discount  and  Interest,  $300  ; Furniture 
and  Fixtures,  $150.  The  items  of  gain  were  : Merchandise,  $750  ; Rent,  $300. 
What  is  the  present  worth  of  the  business?' 

910.  To  find  the  present  worth  of  a business  when  the  resources 
and  liabilities  are  given. 

Example. — The  resources  of  a business  are  Cash,  $1500  ; Merchandise, 
$2420;  Bills  Receivable,  $3100 ; Accounts  Receivable,  $850 ; and  the  liabilities 
are  Bills  Payable,  $1500  ; Accounts  Payable,  $2400.  What  is  the  present  worth? 


Resources. 

Liabilities. 

Cash 

$1500.00 

Bills  Payable  $1500.00 

Merchandise 

2420.00 

Accounts  Payable  2400.00 

Bills  Receivable 

3100.00 

Present  Worth  3970.00 

Accounts  Receivable 

850.00 

$7870.00 

$7870.00 

911.  Rule. — Find  the  difference  between  the  resources  and  the  liabilities.  The 
excess  of  resources  over  liabilities  shows  the  present  worth  ; the  deficiency  of  resources 
below  liabilities  shows  the  insolvency. 

WRITTEN  PROBLEM 

912.  The  resources  of  a business  are  Merchandise,  $4340  ; Cash,  $2810 ; 
Bills  Receivable,  $1300 ; Accounts  Receivable,  $1500  ; and  the  liabilities  are 
Bills  Payable,  $8420;  Accounts  Payable,  $3200.  What  is  the  condition  of  the 
business  ? 


PARTNERSHIP  SETTLEMENTS 


287 


913.  To  distribute  gain  or  loss  when  periods  of  investment  are  equal. 

That  is,  to  find  the  amount  of  gain  to  which  each  partner  is  entitled,  or  the 
amount  of  loss  each  partner  is  to  sustain,  when  the  division  of  loss  or  gain  is  to 
be  made  in  proportion  to  investments,  and  these  investments  are  made  for  the 
same  length  of  time. 

Example. — A,  B and  C form  a partnership,  agreeing  to  share  the  gains  or 
losses  in  proportion  to  their  investments.  A invests  $9500;  B,  $12600;  C,  $7300. 
They  gain  $6300.  Find  each  partner’s  share. 


Operation  by  Fractional  Method 


A $9500 
B 12600 
C 7300 

Total  $29400 

Or,  150 

m 
2 100 
95X0400 
204 
9$ 

14 


vWA  of  $6300 
UrU  of  $6300 
^VWo  of  $6300 


14250 

= 2035.71 
7 


150 

30(3 

900 

73X0400 10950 

~200  7 

4? 

14 


$2035.71  A’s  gain 
$2700.00  B’s  gain 
$1564.29  C’s  gain 


150 
18  900 

120X6300 
g ' - = 2700 

294 

02 

4 


1564.29 


Operation  by  Proportion 

The  ratio  of  the  total  investment  to  each  man’s  investment  equals  the  ratio  of  the  total  gain  to 
each  man’s  gain. 

$29400  : $9500  : : $6300  : A’s  gain  = $2035.71 

$29400  : $12600  : : $6300  : B’s  gain  = $2700.00 

$29400  : $7300  : : $6300  : C’s  gain  = $1564.29 


Operation  by  Percentage  Method 

$6300-?-$29400  = 21.4285% 
$9500X21.4285  = $2035.71 
$12600X21.4285  = $2700.00 
$7300X21.4285  = $1564.29 


288 


PARTNERSHIP  SETTLEMENTS 


914.  To  distribute  gain  or  loss  when  periods  of  investment  are 
unequal. 

That  is,  to  find  the  amount  of  gain  to  which  each  partner  is  entitled,  or  ihe 
amount  of  loss  each  partner  is  to  sustain,  when  the  division  of  loss  or  gain  is  to 
be  made  in  proportion  to  the  investments,  and  these  investments  are  made  for 
different  lengths  of  time. 

Example. — A and  B form  a partnership,  agreeing  to  divide  the  losses  or 
gains  in  proportion  to  investments. 

On  January  1,  A invests  $5200  and  B invests  $4800.  A,  on  May  1, 
invests  $2200  additional,  and  on  July  1,  $3000.  He  withdraws  on  August  1, 
$2800  and  on  November  1,  $3300.  B invests  on  March  1,  $2600,  and  on  June  1, 
he  withdraws  $3000.  They  gain  $3860.  Find  each  partner’s  share. 


Dr. 

A. 

Cr. 

Aug. 

1 

Cash 

2800 

: Jan. 

1 

Cash 

5200 

Nov. 

1 

Cash 

3300 

May 

1 

Cash 

2200 

| July 

1 

Cash 

3000 

2800X5  = 14000 

5200X12  = 62400 

3300X2=  6600 

2200  X 8 = 17600 

20600 

3000  X 6 = 18000 

98000 

20600 

A’s  net  investment  for  1 mo.=  77400 

Dr. 

I 

Cr. 

June 

1 

Cash 

3000 

Jan. 

1 

Cash 

4800 

Mar. 

1 

Cash 

2600 

3000X7=21000 

4800X12=  57600 

2600x10  = 26000 

83600 

21000 

B’s  net  investment  for  1 mo.=  62600 
$77400  A’s  net  investment  for  1 mo. 

62600  B’s  “ “ “ 1 “ 

$140000  total  investment  for  1 mo. 

$140000  : $77400  : : $3860  : A’s  gain  = $2134.03 
$140000  : $62600  : : $3860  : B’s  gain  = $1725.97 

Explanation. — A’s  investment  on  January  1,  $5200,  was  invested  for  12  months.  An  invest- 
ment of  $5200  for  12  months  is  equal  to  an  investment  of  $62400  for  1 month.  The  $2200  was  invested 
for  8 months,  which  is  equal  to  an  investment  of  $17600  for  1 month.  The  $3000  was  invested  for  6 


PARTNERSHIP  SETTLEMENTS 


289 


months,  which  is  equal  to  an  investment  of  $18000  for  1 month.  A’s  total  investment  is  equal  to  an 
investment  of  $98000  for  1 month.  Considering  his  withdrawals,  we  find  the  $2800  to  have  been  out  of 
the  business  for  5 months.  A’s  withdrawal  of  $2800  for  5 months  is  equal  to  a withdrawal  of  $14000 
for  1 month  ; and  the  $3300  for  2 months  is  equal  to  $6600  for  1 month.  His  total  withdrawals  are 
equal  to  a withdrawal  of  $20600  for  1 month.  Subtracting  the  total  withdrawal  for  1 month  from  the 
total  investment  for  1 mouth,  he  has  a net  investment  for  1 month  of  $77400. 

We  proceed  in  the  same  way  with  B’s  account,  and  find  that  he  has  a net  investment  for  1 month 
of  $62600.  Now  these  net  investments  for  one  month  will  form  a true  basis  for  the  division  of  the  gain. 
Taking  the  sum  of  these  investments  as  the  first  term  of  a proportion,  A’s  investment  for  the  second 
term,  the  total  gain  for  the  third,  we  find  the  fourth  term,  or  A’s  gain,  to  be  $2134.03.  In  the  same 
manner  we  find  B's  gain  to  be  $1725.97. 

915.  Rule. — Multiply  each  investment  by  the  time  it  was  invested,  and  each 
withdrawal  by  the  time  it  was  withdrawn.  The  difference  between  the  sums  of  these 
products  is  the  investment,  which  forms  a basis  for  the  division  of  gain  or  loss.  Then 
by  proportion  find  gain  or  loss  of  each  partner. 

916.  Formula  for  determining  each  partner’s  share  of  loss  or  gain. 

Whole  investment : Each  partner’s  investment  : : Whole  gain  : Each  partner’s 
share  of  gain. 

917.  To  find  each  partner’s  capital. 

That  is,  to  find  each  partner’s  capital  at  time  of  closing  the  accounts,  when 
interest  is  allowed  on  deposits  and  charged  on  withdrawals,  the  gain  or  loss  to  be 
divided  equally,  or  in  a given  proportion. 

Example. — A and  B form  a partnership,  agreeing  to  share  equally  the 
losses  or  gains.  It  is  also  agreed  that  each  partner  is  to  receive  interest  on  his 
investment  at  8%.  The  following  is  a statement  of  their  investments  and 
withdrawals.  They  gain  $4250.  Close  their  accounts. 


Dr. 


A. 


Or. 


1907 

Apr. 

9 

Cash 

1800 

00 

Nov. 

6 

Cash 

600 

00 

Dec. 

31 

Balance 

8738 

19 

11138 

19 

Int.  on  $1800  for  266  da.  --  $106.40 
“ “ 600  “ 55  “ = 7 33 


$113.73 


1907 

Jan. 

1 

Cash 

2500 

00 

Mar. 

16 

Cash 

3600 

00 

Sept. 

9 

Cash 

2900 

00 

Dec. 

31 

Interest 

391 

09 

(6 

31 

Gain 

1747 

10 

1908 

11138 

19 

Jan. 

1 

Balance 

1 

Oo 

CO 

Co 

19 

Int.  on  $2500  for  1 yr.  = $200.00 
“ “ 3600  “ 290  da.  = 232.00 

“ “ 2900  “ 113  “ = 72.82 

$504.82 

113.73 

A's  net  credit  interest,  $391.09 


290 


PARTNERSHIP  SETTLEMENTS 


Dr. 


B. 


Or. 


1907 

Mar. 

6 

Cash 

900  00 

Sept. 

19 

Cash 

1200 

00 

Dec. 

31 

Balance 

8fll 

81 

1051181 

u 

Int.  of  $900  for  300  da.  = $60.00 
“ “ 1200  “ 103  “ = 27.47 


$87.47 


$391.09  + $364.71  = $755.80. 
$4250  00  — $755.80  - $3494.20. 


1907 

Jan. 

1 

Cash 

3200  00 

Apr. 

9 

Cash 

1600  00 

Aug. 

26 

Cash 

3600  00 

Dec. 

31 

Interest 

364  71 

i i 

31 

Gain 

1747  10 

1051181 

1908 

Jan. 

1 

Balance 

841181 

Int. 

of  $3200  for  1 yr. 

= $256.00 

U 

U 

1600  “ 266  da. 

= 94  58 

U 

:c 

3600  “ 127  “ 

- 101.60 

$452.18 

87.47 

B’s  net  credit  interest,  $364.71 


$3494.20  h-  2 = $1747.10,  each  partner’s  share  of  the  gain. 


918.  Rule. — Find  the  interest  on  the  investments  for  the  time  for  which  they  were 
made.  Find  also  the  interest  on  the  withdrawals  for  the  time  they  were  withdrawn. 
The  difference  between  the  sums  of  these  interest  items  is  the  interest  with  which  the 
account  should  be  either  credited  or  debited.  Then  divide  the  remainder  of  the  gain  or 
loss.  Enter  this  on  the  proper  side  of  the  account  and  find  the  balance. 

WRITTEN  PROBLEMS 


919.  1.  A and  B form  a partnership,  agreeing  to  divide  the  gains  or  losses 
in  proportion  to  investments.  A invests  $7500  and  B invests  $9600.  Divide  a 
gain  of  $3750. 

2.  Three  men  rent  a pasture  for  three  months  for  $96.  One  puts  into  it 
14  head  of  cattle,  the  second  22  head,  the  third  19  head.  What  should  each  be 
required  to  pay  ? 

3.  A,  B and  C enter  into  partnership.  A puts  in  $1000  for  8 months  : B. 
$1200  for  10  months;  C,  $800  for  12  months.  They  gain  $1500.  What  was  the 
share  of  each  ? 

J.  A set  of  books  shows  the  following  results:  Loss  and  Gain,  Dr.,  $7895.00, 
Cr.,  $9873  21;  A’s  Capital,  Cr.,  $3175.29,  withdrawals,  $S46.71  : B’s  Capital,  Cr., 
$6295.35,  withdrawals,  $1237.18.  B is  to  have  a salary  of  $2000;  an  interest 
account  is  to  be  kept,  to  adjust  the  differences  in  capital.  After  crediting  salary 
and  one  year’s  interest  at  6%  on  investments,  the  balance  is  shared  by  A and  B 
equally.  AVhat  are  the  proper  balances  for  A and  B to  commence  the  new 
period  ? 

5.  Loss  and  Gain,  Dr.,  $38967.81,  Cr.,  $92865.28  ; A’s  account,  Dr.,  $4567.28, 
Cr.,  $24758.62  ; B’s  account,  Dr.,  $3967.19,  Cr.,  $16969.56.  Before  settling,  credit 
A’s  salary,  $2500  ; credit  B’s  salary,  $4500  ; also  credit  A and  B interest  at 


PARTNERSHIP  SETTLEMENTS 


291 


6 per  cent,  on  the  original  investment;  then  close  Loss  and  Gain,  giving  each 
partner  half  of  net  gain.  State  each  partner’s  balance  at  closing. 

6.  A set  of  books  shows  net  gain  of  $8937.62;  A’s  investment,  $2500.00; 
B’s  investment,  $3900.00  ; A drew  $3016.28  ; B drew  $9173.21.  Interest  is  to  be 
allowed  on  the  investment  of  each  partner,  but  is  not  to  be  charged  on  with- 
drawals, and  the  balance  of  net  gain  is  to  be  evenly  divided.  What  is  the  balance 
of  each  partner’s  account? 

7.  Loss  and  gain,  credit  in  excess  of  debit,  $9346.21  ; Ames’s  debits,  $1300.00, 
credits,  $15265.75;  Bell’s  debits,  $900.00,  credits,  $6875.94;  Carr’s  credits, 
$10000.00.  The  agreement  allows  Bell  $1000  for  services,  and  each  partner  is  to 
be  paid  with  interest  before  the  profits  are  divided.  Bell  is  to  pay  12  per  cent, 
interest  to  Carr  for  the  use  of  his  money ; Carr  is  not  a partner,  but  loans  money 
to  the  firm  of  Ames  & Bell,  at  the  expense  of  Bell  alone  ; what  will  be  the  balance 
of  Ames’s  and  Bell’s  accounts  after  settlement  ? 

8.  Close  books  containing  the  following:  Net  gain,  $8937.65.  Partners’ 
capital;  A — Cr.,  $9654.62  ; B — Cr.,  $6978.78 ; C — Cr.,  $11964.65.  Partners  were 
debited  as  follows : A— Dr.,  $2245.67 ; B— Dr.,  $1463.79  ; C— Dr.,  $1296.55.  The 
articles  of  agreement  provide  that  the  partners  shall  receive  salary  for  their 
services  : A,  $2000  ; B,  $1500  ; and  C,  $3000.  The\r  shall  have  interest  on  capital, 
but  no  interest  is  charged  on  withdrawals;  the  balance,  gain  or  loss,  shall  then 
be  equally  divided. 

9.  A,  B and  C traded  in  company  ; A put  in  $1  as  often  as  B put  in  $3, 
and  B put  in  $2  as  often  as  C put  in  $5.  B’s  money  was  in  twice  as  long  as  C’s 
and  A’s  twice  as  long  as  B’s ; they  gained  $5250;  how  much  was  each  man’s 
share  of  the  gain  ? 

10.  Armor  and  Baker  engaged  in  a partnership  for  four  years ; Armor  put 
in  $6000  and  Baker  $8000.  At  the  close  of  the  second  year  Armor  took  out 
$2000,  and  Baker  putin  $2000;  at  the  close  of  the  fourth  year  they  divided 
$8890  as  net  gain.  What  was  the  share  of  each  ? 

11.  A certain  ledger,  after  all  the  work  was  posted  and  the  accounts  closed, 
showed  the  following  results  : Loss  and  Gain,  debits,  $2829.46,  credits,  $18765.40  ; 
A.  Bard,  debits,  $2526.89,  credits,  $26345.48;  C.  Dash,  debits,  $3687.95,  credits, 
$19875.69.  The  articles  of  copartnership  require  an  interest  account  kept  with 
the  partners,  crediting  each  with  interest  on  capital  invested,  but  not  charging 
interest  on  withdrawals ; C.  Dash  is  to  be  credited  $1000  a year,  because  he  is  the 
outside  partner,  and  is  subjected  to  personal  expenses  on  that  account.  Com- 
plete the  closing  of  the  ledger  and  determine  the  balance  to  the  credit  of  each 
partner. 

1 2.  Determine  the  balances  to  the  partners’  credit  in  a set  of  books  exhibit- 
ing results  as  follows  ; Loss  and  Gain,  net  credit,  $12365.92  ; Black’s  capital  at 
beginning,  $10000.00  ; Green’s  capital  at  beginning,  $5000.00.  Black  drew  $3305  ; 
Green  drew  $2976;  Green  is  to  have  a salary  of  $3000,  and  each  partner  is  to 
have  interest  at  the  rate  of  5 per  cent,  on  his  capital.  Gains  and  losses  shared 
equally. 


292 


PARTNERSHIP  SETTLEMENTS 


13.  Loss  and  Gain  account,  Dr.,  $3567.89,  Cr.,  $19656  25.  Hyde’s  personal 
account,  Dr.,  $3467.89,  Hyde’s  capital  account,  Cr.,  $15624.00 ; Jones’s  personal 
account,  Dr.,  $4275.00,  Jones’s  capital  account,  Cr.,  $8767.49.  Jones  receives  a 
salary  of  $2500.  Interest  is  allowed  on  capital  and  the  balance  is  equally 
divided.  Wliat  is  the  balance  to  the  credit  of  each  partner  at  the  beginning 
of  the  new  year? 

Ilf..  Lome  and  Sage  engage  in  trade.  Lome  puts  in  $5000,  and  at  the  end 
of  four  months  takes  out  a certain  sum.  Sage  puts  in  $2500,  and  at  the  end  of 
five  months  puts  in  $3000  more.  At  the  end  of  the  year  Lome’s  gain  is 
$1066§,  and  Sage’s  is  $1333J.  What  sum  did  Lome  take  out  at  the  end  of  four 
months  ? 

15.  Three  men  take  an  interest  in  a coal  mine.  B invests  his  capital  for 
four  months,  and  claims  one  tenth  of  the  profits  ; C’s  capital  is  in  eight  months; 
D invests  $6000  for  six  months  and  claims  two-fifths  of  the  profits;  how  much 
did  B and  C put  in?  Losses  and  gains  to  be  divided  in  proportion  to  invest- 
ments. 

16.  X,  Y and  Z entered  into  business  as  partners,  each  putting  in  $5000  as 
capital.  At  the  end  of  two  years  X took  out  $1000,  Y $2000,  and  Z $3000.  At 
the  end  of  the  fourth  year  they  closed  the  business  with  a loss  of  $3600.  What 
was  the  loss  of  each  ? Losses  and  gains  to  be  divided  in  proportion  to  invest- 
ments. 

17.  Close  a set  of  books  which  shows  the  following  results,  and  state  each 
partner’s  balance.  Net  credit  to  Loss  and  Gain  account,  $25000.  White,  Dr., 
$1800,  Cr.,  $12000.  Redd,  Dr.,  $2500,  Cr.,  $10000  ; Redd  to  be  allowed  $1000  for 
services,  and  each  partner  to  be  allowed  interest  on  his  capital,  and  the  net  gain 
to  be  divided  equally. 

18.  The  Loss  and  Gain  account  in  a set  of  books  is  credited  $31684.89. 
Abel’s  account  (senior  partner),  debited  $2937.63.  credited  $60758.93.  Bearer’s 
account  ( junior  partner),  debited  $925.76,  credited  $17896.49.  Determine  and 
credit  the  interest,  ignoring  withdrawals  of  partners.  Credit  each  partner  $3000 
for  services.  Divide  the  net  profits  equally  and  bring  down  the  new  balances  to 
the  credit  of  partners.  What  are  they? 

19.  Two  men  purchase  a house  which  rents  for  $420  a year.  One  of  them 
pays  $2200  of  the  purchase  money.  The  other  receives  $295  as  bis  share  of  the 
rent.  Find  the  amount  of  the  purchase  money  the  second  one  pays.  If  the  total 
amount  paid  for  taxes,  repairs,  etc.,  is  $105,  how  much  of  this  should  each  pay '? 

20.  Three  boys,  A,  B and  C,  buy  a bicycle,  A paying  $32,  B $48  and  C $20. 
They  agree  that  each  shall  have  the  use  of  it,  in  even7  week  of  six  days,  10  hours 
each,  for  a time  proportionate  to  their  payments.  If  A begins  the  use  of  it  at  six 
o’clock  Monday  morning,  when  should  B receive  it?  When  should  C receive  it‘.' 

21.  A,  B and  C form  a partnership,  agreeing  to  share  gains  or  losses  in 
proportion  to  investments.  A invests  $4600.  They  gain  $3800.  B receives  as 
his  share  of  the  gain  $1500  and  C receives  4 of  the  gain.  Find  B s and  C’s 
investments. 


PARTNERSHIP  SETTLEMENTS 


293 


22.  A and  B form  a partnership,  agreeing  to  share  losses  and  gains  in  pro- 

portion to  average  net  investments.  On  January  1,  1908,  A invests  $3200,  and 
B on  the  same  date  invests  $2650.  A on  March  1 invests  $1600  and  on  Sep- 
tember 1,  $2500,  and  withdraws  on  May  1,  $1600  and  on  October  1,  $550.  B 

invests  on  April  1,  $1800,  and  on  August  1,  $1950,  and  withdraws  on  June  1, 

$1000  and  on  December  1,  $900.  The  present  worth  at  the  close  of  the  year  is 
$12500.  How  much  is  due  each  partner? 

23.  A and  B form  a partnership,  agreeing  to  share  gains  and  losses  in  pro- 
portion to  average  net  investments.  A on  January  1 invests  $4800,  and  B on 
the  same  date  invests  $5300.  On  May  1,  A invests  $2200,  and  on  August  1 
withdraws  $1200.  B on  April  1 invests  $4300,  and  on  September  1 he  with- 
draws $1900.  They  take  in  C as  a partner  on  July  1,  and  lie  invests  a sufficient 
amount  to  entitle  him  to  -l- of  the  gain.  Find  A’s  and  B’s  gain  and  C’s  invest- 
ment. They  gain  $6200. 

21/..  A and  B form  a partnership  on  January  1,  1908,  agreeing  to  share  the 
gains  and  losses  in  proportion  to  their  average  net  investments.  A on  January 
1 invests  $2600,  and  B $4800.  A invests  on  July  6,  $2800,  and  withdraws 

on  September  1,  $1200.  B,  on  September  1,  withdraws  such  an  amount  as  will 

make  his  average  investment  sufficient  to  entitle  him  to  f of  the  gain,  and  C on 
June  1 invests  a sufficient  amount  for  the  rest  of  the  year  to  entitle  him  to  J of 
the  profits.  Find  B’s  withdrawal,  A’s  gain,  and  C’s  investment.  They  gain 
$5800. 

25.  A and  B form  a partnership,  agreeing  to  share  losses  and  gains  in  pro- 
portion to  average  net  investments.  A on  January  1 invests  $5500,  and  B on 
the  same  date  invests  $4200.  A withdraws  on  June  1,  $1200,  and  he  invests 
on  August  1,  $1400.  B withdraws  on  September  1,  $1500,  and  invests  on 
November  1,  $2500.  C is  admitted  to  the  partnership  and  invests  $3800  for 
a time  sufficient  to  entitle  him  to  £ of  the  gain.  Find  the  date  on  which  C made 
his  investment. 

26.  Three  men,  A,  B and  C,  rent  a pasture  for  4 months,  for  which  they 
agree  to  pay  $250.  A puts  in  25  cattle  for  2 months,  and  32  additional  for  the 
remaining  2 months.  B puts  in  18  cattle  for  1 month,  and  for  the  remaining 
three  months  has  in  26.  C puts  in  28  for  3 months,  and  for  the  remaining 
month  has  in  34.  Find  the  amount  each  should  pay. 

27.  John  Rankin,  of  Texas,  and  James  Moore,  of  Chicago,  agree  to  form  a 
partnership  for  carrying  on  a trade  in  cattle.  To  begin  the  business,  Moore 
sends  Rankin  $12000.  Rankin  buys  cattle  during  the  year  to  the  amount  of 
$22500,  and  ships  to  Moore  cattle  of  which  Moore  sells  to  the  amount  of  $9650, 
and  Rankin  sells  to  the  amount  of  $11920.  They  agree  to  dissolve  partnership, 
and  an  examination  of  their  books  shows  that  Rankin  has  paid  for  expenses 
during  the  year  $922  and  Moore  has  paid  $1150.  Rankin  has  on  hand  at  the 
time  of  settlement  cattle  valued  at  $5020,  and  Moore  has  on  hand  cattle  valued 
at  $6211.  It  is  agreed  between  them  that  each  will  take  the  cattle  he  has  on 
hand  at  the  price  given.  Find  their  gain  or  loss.  Find  also  which  one  owes 
the  other,  and  how  much. 


294 


PARTNERSHIP  SETTLEMENTS 


28.  A,  B,  C and  D engage  the  services  of  a teacher  for  9 months,  and  agree 
to  pay  him  $1200  ; each  is  to  pay  in  proportion  to  the  number  of  children  sent. 
A sends  three  children  for  180  days;  B sends  one  child  for  180  days  and  one  for 
160  days;  C sends  three  children  for  170  days  and  two  for  130  days  ; and  D 
sends  two  children  for  180  days,  one  for  170  days  and  two  for  140  days.  Find 
the  amount  due  from  each. 

29.  X,  Y and  Z agree  to  do  a piece  of  work,  for  which  they  are  to  receive 
$970;  each  one  is  to  receive  the  same  amount  per  hour.  X works  32  days  of  9 
hours  each;  Y works  18  days  of  10  hours  and  22  days  of  9 hours;  Z works  12 
days  of  8 hours  and  18  days  of  10  hours.  Divide  the  sum  among  them. 

30.  Two  men,  M and  N,  hired  a team  to  go  a distance  of  10  miles  and 
return  ; the  charge  for  the  team  was  $8.  On  arriving  at  their  destination  M 
proposed  to  return  by  another  conveyance,  to  which  X agreed.  They  also  agreed 
that  M should  only  be  required  to  pay  in  proportion  to  the  number  of  miles  he 
had  ridden.  Find  what  each  must  pay. 

31.  A man  had  two  sons  and  four  daughters.  To  the  3roungest  son  he  gave 
4 of  three  times  $2800,  which  was  § of  the  share  of  the  older  son,  and  what  the 
younger  son  received  was  J of  f of  the  entire  estate.  The  balance  of  the  estate 
was  divided  among  his  four  daughters  in  reciprocal  proportion  to  their  ages — 
12,  16,  18,  21.  Find  each  daughter’s  share. 

32.  A father  settled  his  estate,  which  was  valued  at  $39000,  upon  his  six 
children  in  the  following  proportion:  William  John  l,  Ann  ^ , Mary  4,  Samuel 
1 and  Howard  What  sum  should  each  receive? 

33.  William  Reid  and  John  Day  agree  to  form  an  equal  partnership.  They 
buy  a planing  mill,  the  price  of  which  is  $15000,  $7000  of  which  Reid  pays  and 
the  balance  is  paid  by  Day.  They  buy  rough  lumber  to  the  amount  of  $3900, 
half  of  which  is  paid  by  each.  Reid  pays  during  the  year  for  wages  and  other 
expenses  $3950,  and  Day  pays  $2620.  Their  sales  of  finished  product  during  the 
year  amount  to  $7320,  and  they  make  an  additional  purchase  of  rough  lumber 
to  the  amount  of  $1850.  Their  taxes  and  insurance  during  the  year  amount 
to  $310.  At  the  time  of  settlement,  at  the  close  of  the  year,  they  have  on  hand 
rough  lumber  to  the  amount  of  $920  and  finished  product  to  the  amouut 
of  $1750.  Day  is  to  receive  $1200  for  conducting  the  business,  and  Reid 
$1200  for  managing  the  mill.  It  is  required  to  know,  first,  their  gain  or  loss  ; 
second,  in  case  Reid  wishes  to  sell  out,  how  much  Day  should  pay  him  for  his 
interest. 

34-  Three  families  agree  to  support  a private  tutor  for  their  children,  and 
for  this  purpose  rent  a room  at  an  expense  of  $250.  They  agree  to  pay  the 
teacher  $1000  for  300  days’  service,  6 hours  each,  and  it  is  agreed  among  them 
that  they  shall  pay  in  proportion  to  the  number  of  children  sent  and  the  hours 
they  are  taught.  The  first  family  sends  five  children  for  the  full  time  ; the  second, 
two  children  for  300  days  and  four  children  for  210  days,  and  the  third  family  is 


PARTNERSHIP  SETTLEMENTS 


295 


required  to  pay  ^ of  the  expense ; this  family  sent  four  children  for  the  same 
length  of  time.  It  is  required  to  know  how  many  days  the  children  of  the  third 
family  were  sent. 

35.  In  a certain  school  district  there  are  seven  families.  The  first  is  located 
1 mile  from  the  schoolhouse;  the  second,  1^  miles ; the  third,  2J  miles;  the 
fourth,  2|-  miles;  the  fifth,  2f  miles;  the  sixth,  3J  miles  ; the  seventh,  4 miles. 
The  total  amount  of  tax  for  school  purposes  to  be  raised  by  these  families  is  $170, 
which  it  is  agreed  they  shall  pay  in  reciprocal  proportion  to  the  distance  they  are 
located  from  the  school.  Find  the  amount  paid  by  each. 

36.  Reese,  Wood,  Harris  and  Mills  form  a partnership — Reese  4,  Wood 
Harris  Mills  | ; the  remainder  of  the  gain  is  left  in  the  business.  To  secure 
these  proportional  shares  of  profits,  it  is  agreed  that  each  partner  shall  invest 
$12500.  Harris,  however,  invested  only  $9500,  and  Mills  $10200.  The  net  gain 
amounted  to  $1720.  Since  Harris  and  Mills  did  not  invest  the  required  amount, 
determine  each  one’s  share  of  the  gain. 

37.  A,  B and  C form  a partnership,  agreeing  to  share  losses  and  gains 
equally.  At  the  close  of  the  year  it  is  found  that  A has  to  his  credit  $9200,  that 
B has  overdrawn  his  account  $1500,  and  that  C has  overdrawn  his  account  $2150. 
There  being  no  other  resources  or  liabilities,  determine  an  equitable  dissolution. 

38.  A and  B form  an  equal  partnership.  At  the  close  of  the  year  A’s 
account  is  overdrawn  $2900,  and  B’s  $2300.  How  may  they  settle? 

39.  A and  B are  doctors,  and  agree  to  combine  their  interests  and  share 
equally  in  the  net  amount  made  during  a year.  A owns  the  office  which  they 
use,  and  values  the  yearly  rental  at  $750.  B supplies  the  horses  and  carriages 
necessary  to  the  business  and  pays  $1000  for  their  care  and  keep;  their  value  is 
$900.  He  charges  15%  of  this  value  for  their  use.  A collects  during  the  year 
$4225,  and  B $5006.  A’s  incidental  expenses  connected  with  the  business  are 
$112.50,  and  B’s  are  $212.75.  Determine  each  one’s  share  of  the  receipts  and 
how  they  shall  settle. 

Jf.0.  A company  engaged  an  agent  to  do  business  for  them  for  four  months 
at  a salary  of  $175  a month.  They  shipped  him  merchandise  amounting  to 
$3920.50,  and  sent  him  also  $920  cash.  During  the  four  months  the  agent 
purchased  goods  amounting  to  $2140.50.  At  the  end  of  the  time  the  agent’s  sales 
amounted  to  $3950.82,  and  the  goods  on  hand  amounted  to  $1005.26.  Find  gain 
or  loss.  Find  also  the  amount  due  the  company  at  the  end  of  that  time. 

Ifl.  Lippincott,  Kehm  and  Cressman  are  partners  in  business,  and  at  the 
close  of  the  year  their  books  show  that  Lippincott  has  overdrawn  his  account 
$1050  and  Kehm  has  overdrawn  his  $320.  The  firm  assumes  a private  debt  of 
$960  for  Cressman.  When  their  books  were  last  closed,  each  partner  had  an 
equal  sum  to  his  credit.  The  resources  at  the  present  time  are:  cash,  $10250 ; 
bills  receivable,  $4100  ; merchandise,  $9285;  book  accounts,  $726.  Their  liabili- 
ties are:  bills  payable,  $9175;  interest,  $422;  sundry  accounts,  $7005.  What  is 
■each  partner’s  capital  at  closing,  the  gain  being  $5900? 


296 


PARTNERSHIP  SETTLEMENTS 


40.  Mariner  and  Sailor  are  joint  owners  of  a vessel  which  cost  §80000,  of 
which  Mariner  paid  $50000  and  Sailor  $30000.  They  sell  one-third  interest 
to  Merchant  for  $50000,  each  retaining  one-third  interest.  What  part  of  the 
amount  paid  by  Merchant  should  each  receive? 

43.  The  firm  of  Sharp  & Dull  have  just  closed  their  books,  which  show  a 
gain  of  $550.  A statement  of  resources  and  liabilities  shows  the  following : 
Cash,  $8000;  Merchandise,  $13000  ; Real  Estate,  $5000  ; Bills  Receivable,  $3500; 
Accounts  Receivable,  $6750  ; Bills  Payable,  $8000;  Accounts  Payable,  $15200. 
What  was  Sharp’s  investment,  Dull  having  invested  $2500  ? 

44-  The  firm  of  Platt  & Son  find  upon  closing  their  books  a net  loss  of 
$8700.  A statement  of  resources  and  liabilities  shows  Cash,  $1800;  Merchan- 
dise, $11000;  Bills  Receivable,  $500  ; Accounts  Receivable,  $1500;  Bills  Par- 
able, $4000 ; Accounts  Payable,  $7500.  What  was  the  net  capital  of  Platt  & 
Son  at  the  opening  of  the  season  ? 

45.  Henry  Jones  has  his  business  incorporated.  His  assets  are  Cash,  $9000  ; 
Real  Estate,  $15000 ; Mdse.,  $10000 ; Bills  Receivable,  $560;  Accounts  Receiv- 
able, $8950.  He  owes  Bills  Payable,  $1200;  and  Accounts  Payable,  $3500. 
The  business  is  incorporated  under  the  title  of  the  “ Linen  Manufacturing 
Company,”  with  a capital  of  $200000,  divided  into  2000  shares  at  §100  each. 
Henry  Jones  takes  500  shares  for  his  share;  George  Long  takes  500  shares: 
William  Day  takes  500  shares;  William  Smith  takes  400  shares;  Leonard  Hay 
receives  100  shares  for  his  services  as  attorney.  William  Day  buys  his  shares 
at  par  and  pays  one-half  cash  and  note  for  balance.  George  Long  buys  his 
at  95  and  pays  cash  for  them.  William  Smith  buys  at  105,  paying  one-half  cash 
and  turns  over  real  estate  for  balance.  What  does  each  pay  for  his  interest  in 
the  corporation?  What  is  the  real  value  of  each  one’s  interest? 

46.  A merchant  is  doing  business  and  keeps  his  books  by  single  entry.  He 
began  with  Cash  $10000.  He  bought  Mdse,  from  A,  $7000;  B.  $3000 ; C,  $4000; 
D,  $1000.  Paid  A $4000  and  note  $3000.  Sold  E $5000  and  paid  B in  full. 
Sold  F $7500,  gave  C on  account  $3000  and  paid  D in  full.  Sold  G S5300. 
Received  from  E $4000.  Sold  H $4200.  Received  from  F cash  $5000  and  note 
for  $2500,  and  from  G cash  $3000  and  note  $1300.  Received  from  H cash  $1000 
and  note  $1200.  Wishing  to  change  his  books  to  double  entry,  he  took  an 
account  of  stock  and  made  the  change.  Inventory,  Mdse.,  $2500.  What  is  the 
present  worth  of  the  business? 

47.  A and  B are  partners,  A having  f interest  and  B J interest  in  gains  and 
losses.  They  have  disposed  of  everything  belonging  to  the  business  and  have 
on  hand  $5000  cash.  They  wish  to  settle  with  each  other.  A’s  account  shows 
a net  credit  of  $6000,  while  B’s  account  shows  a net  debit  of  $3500.  What  is 
the  final  and  equitable  settlement  between  them? 

4.8.  A Philadelphia  firm  opened  a branch  store  in  Bethlehem,  Pa.,  and  put 
J.  M.  Fayth  in  charge.  He  is  to  receive  5%  commission  on  all  sales  and 


PARTNERSHIP  SETTLEMENTS 


207 


incomes  of  the  business.  He  employs  an  assistant  at  a salary  of  $1500,  which 
is  paid  by  himself.  The  firm  supplied  the  branch  as  follows  : Merchandise, 

$15000;  Cash,  $5000,  During  the  year  Faylh  bought  for  the  branch  $175325 
and  the  sales  were  $222900.  He  received  a consignment  of  goods  invoiced  at 
$1200,  from  Wilmington,  Del.,  upon  which  he  paid  freight  $100  and  storage  $75. 
The  consignment  was  disposed  of  at  an  advance  of  10%  above  invoice  value.  A 
commission  of  3 % was  received  and  proceeds  remitted.  The  running  expenses 
of  the  business,  rent,  etc.,  outside  of  assistant’s  salary,  were  $2300  ; these  come 
out  of  the  business  and  have  been  paid.  The  Merchandise  inventory  is  $4500, 
and  there  is  $6500  cash  on  hand.  What  is  the  result  of  the  business?  What 
has  been  Fayth’s  income  from  the  business? 

Ifd.  A and  B form  a copartnership,  agreeing  to  share  gains  or  losses  in  pro- 
portion to  average  net  investment.  On  January  1 each  invests  $6000.  On 
February  1 A invests  $2000  additional,  and  B withdraws  $500.  On  March  1 
A withdraws  $750.  On  April  1 C is  admitted  to  the  firm,  and  invests  $10000. 
On  May  1 B invests  $1000  additional.  On  June  1 A withdraws  $500.  On 
July  1 D is  admitted  to  the  firm,  and  invests  $7500.  On  August  1 C with- 
draws $2500.  On  September  1 B withdraws  $400.  On  October  1 D withdraws 
$500.  On  November  1 A withdraws  $750,  B withdraws  $100,  C withdraws 
$1500,  and  D withdraws  $1000,  leaving  their  investments  equal  for  the  remain- 
der of  the  year. 

On  December  31  their  books  show  the  following  resources  and  liabilities : 
Cash,  $10000;  Mdse.,  $13000 ; Bills  Rec.,  $5000 ; Accts.  Rec.,  $7650;  Inventory 
Mdse.,  $6000  ; Bills  Pay.,  $5000  ; Rent  due  by  us,  $1800 ; Accts.  Pay.,  $4000. 
Allow  interest  at  6%  on  all  investments  and  charge  interest  on  all  withdrawals. 
Find  net  gain  of  each,  and  find  present  worth  of  each  partner. 

50.  June  1,  1905,  Brown  and  Smith  form  a partnership,  agreeing  to  share 
gains  and  losses  in  proportion  to  average  net  investments.  They  have  a joint 
capital  of  $8400,  of  w7hich  Brown  furnished  ^ and  Smith  the  remainder;  Apr.  1, 
1906,  Smith  invested  $1200,  and  Brown  withdrew  $700;  Sept.  1,  1906,  they 
admitted  Gray  to  the  partnership  with  an  investment  of  $4800 ; Jan.  1,  1907, 
each  partner  invested  $1500,  and  on  Jan.  1,  1908,  each  withdrew  $600.  Sept.  1, 
1908,  on  closing  business,  it  was  found  they  had  sustained  a net  loss  of  $2800. 
What  was  each  partner’s  share  of  the  loss  ? 


INVOLUTION  AND  EVOLUTION 

INVOLUTION 


920.  Involution  is  the  process  of  raising  a number  to  any  required  power. 
Any  power  of  a number  is  obtained  by  using  the  number  as  a factor  as  many 
times  as  are  indicated  by  the  exponent,  which  is  a small  figure  written  above 
and  to  the  right  of  the  number.  As, 

31  = 3,  the  first  power  of  3 ; 

32  = 3X3  = 9,  the  second  power,  or  square,  of  3 ; 

33  = 3X3X3  = 27,  the  third  power,  or  cube,  of  3 ; 

34  = 3X3X3X3  = 81,  the  fourth  power  of  3 ; etc. 


921.  The  process  of  raising  numbers  of  two  or  more  figures  to  a required 
power  may  be  formulated  into  a rule  by  separating  the  number  into  its  component 
parts  and  then  indicating  the  multiplication  instead  of  actually  multiplying ; 
thus, 


37  = 30+7 
37  = 30+7 

(30X  7)+72 
302  + (30x7) 

The  square  = 30 2 +2(30X7  ) + 72 

30+7 

(302  x7)  + 2(30X72)  + 73 
30 3 +2(302  X7)+  (30 X72) 

30 3 + 3(30 2 X 7)  + 3(30  X 7 2 ) + 7 3 
which  is  the  cube  of  37. 


or  t+u 

t+u 

(tXu  )+u2 
t2+(tXu  ) 

t2+2(iXu  )+u2 

t+u 

(t2  Xu)+2(tXu2)+u3 
u3+2(t2Xu)+  (tXu2) 

u3  + 3(t2  Xu)+3(tX  u2)+u3 


922.  From  this  we  derive  the  following  principles: 

t.  The  square  of  a number  of  two  figures  (t+u),  equals  tens 2 + 2 times 
tens  X units  + units2. 

2.  The  square  of  a number  of  three  figures  (h  + t + u),  equals  hundreds 2 + 
2 times  hundreds  X tens  + tens 2 + 2 times  ( hundreds  + tens)  X units  + units2. 

3.  The  cube  of  a number  of  two  figures  (t+u),  equals  tens 3 + 3 times  tens 2 
X units  + 3 times  tens  X units 2 + units3. 

4~  The  cube  of  a number  of  three  figures  (h+t  + u),  equals  hundreds3  + 3 
times  hundreds2  X tens  + 3 times  hundreds  X tens 2 + tens3  + 3 times  (hundreds 
+ tens)2  X units  + 3 times  ( hundreds  + tens)  X units2  + units3. 

Note  1. — Higher  powers  may  be  obtained  by  combining  the  lower  ; thus,  44=42X42  ; 45=4SX 
4 2 ; 46  = 43X43,  etc. 

Note  2. — A fraction  is  raised  to  a required  power  by  raising  numerator  and  denominator  sepa- 
rately ; as  (f)2  = f,  etc. 


298 


EVOLUTION 


299 


WRITTEN  EXERCISE 

923.  Find  the  value  of 


1.  142. 

9.  (I)4- 

17.  .092. 

25.  1.6724. 

2.  22 3 . 

10.  (5J)3. 

18.  ,674. 

26.  (2.76J)3 

3.  462. 

11.  (6f)4. 

19.  003 8 . 

27.  9.064. 

4.  293. 

12.  (34) 7 . 

20.  (.17i)5. 

28.  91.673. 

5.  164. 

13.  (22|)5. 

21.  ,274. 

29  22.052. 

6.  175. 

H-  (194)3. 

22.  .086 6 . 

30.  (18f)5. 

7.  9273. 

15.  (26f)2. 

23.  .0721 3. 

31.  (66§)3. 

8.  8424. 

16.  (2i)N 

24-  (.96f)5. 

32.  (33i)4. 

33.  Cube  all 

the  whole  numbers 

from  one  to  six  inclusive 

and  square 

sura  of  those  powers. 

34-  Cube  the  three  highest  digits  and  cube  the  sum  of  those  powers. 

35.  Find  the  value  of  (72+83-f-42+34+56)3. 

36.  Find  the  value  of  (54)6. 


EVOLUTION 

924.  Evolution  is  the  process  of  finding  the  root  of  a number.  'sMU 

925.  A root  of  a number  is  one  of  its  equal  factors.  The  root  is  indicated 

by  the  use  of  the  symbol  -\/  , called  the  radical  sign. 

Note. — The  square  root  of  36  is  6 (]/36  = 6),  because  6 X6  = 36.  The  cube  root  of  216  is  6 
(#/216  = 6),  because  6X6X6  = 216. 

926.  When  the  radical  sign  is  used  alone,  it  indicates  the  square  root. 
When  any  other  root  is  required,  the  figure  indicating  the  root  is  written  in  the 
angle  of  the  radical  sign.  Thus,  i/—  indicates  square  root,  f/  indicates  cube 
root,  i y~  indicates  fourth  root,  etc. 

927.  A perfect  power  is  a number  of  which  the  required  root  can  be 
found  exactly. 

928.  An  imperfect  power  is  a number  whose  root  can  not  be  found 
exactly. 

Square  Root 

929.  Observe  the  following  numbers  and  their  squares  . 

Numbers.  1,  3,  5,  9,  10,  19,  34,  99,  100,  500. 

Squares.  1,  9,  25,  81,  100,  361,  1156,  9801,  10000,  250000. 

From  this  we  learn  : 

1.  That  the  square  of  a number  of  one  figure  contains  one  or  two  figures. 

2.  That  the  square  of  a number  of  two  figures  contains  three  or  four  figures. 

3.  That  the  square  of  a number  of  three  figures  contains  five  or  six  figures. 

4.  In  general,  the  square  of  any  number  contains  twice  as  many  figures  as  the  number,  or  twice 
as  many  less  one. 

5.  That  if  a given  number  be  separated  into  groups  of  two  figures  each,  each  group  or  partial 
group  will  be  represented  by  a figure  in  the  root.  Hence  the  following  rule  : 


300 


EVOLUTION 


930.  .Rule. — Separate  the  number  into  two-figure  groups. 

931.  To  extract  the  square  root  of  a whole  number. 

Example. — Find  the  square  root  of  225. 


2' 27 

/ O 

/ o2  = 

/ O 0 

7 

/ 0 X 2 = 20 

/ 27 

/7  OLuTX 

7 

27 

/ 27 

10 

A 

c 

10 

5 

— 

B £ 

Explanation. — Separate  the  number  into  periods  of  two  figures  each.  This  determines  that 
the  root  contains  2 figures.  The  first  or  left  hand  period  is  2,  and  the  greatest  square  contained  in  it 
is  1 ten  or  10  units.  Place  10  in  the  root-place  at  the  right,  and  subtract  the  square  of  10  or  100 
square  units  from  225,  leaving  125  square  units. 

The  largest  part  of  these  125  square  units  is  in  rectangles  A and  B,  each  of  which  is  ten  units 
long,  or  together  2X10  or  20.  ThisTlivided  into  125  will  give  the  width,  5 units. 

The  remaining  square,  C,  is  as  large  each  way  as  the  width  of  rectangles  A and  B,  5 units.  Add 
this  5 units  to  20  units,  and  we  have  the  length  of  A,  B,  and  C,  25  units.  Multiply  this  length  25  by 
the  width,  5,  and  we  have  the  area,  125  square  units  and  no  remainder. 

The  square  root  is  10+5  or  15  units. 


932.  Rule. — Beginning  at  units,  separate  the  number  into  two-figure  groups. 
Each  period  represents  a figure  in  the  square  root. 

Find  the  largest  number  whose  square  is  contained  in  the  first  or  left-hand  period  ; 
write  it  in  the  first  root-place,  and  put  ciphers  in  the  remaining  root-places.  Subtract 
the  square  of  the  entire  root  from  the  given  number  for  a new  dividend. 

For  a trial  divisor,  multiply  the  root  by  2 ; divide  the  product  into  the  dividend 
and  write  the.quotient  for  the  next  figure  of  the  root.  Add  the  second  part  of  the  root 
to  the  trial  divisor,  and  multiply  the  sum  by  the  root  just  found.  Subtract  for  the  new 
dividend.  Proceed  in  like  manner  until  all  the  periods  have  been  disposed  of. 

933.  To  extract  the  square  root  of  a decimal. 

Example. — Extract  the  square  root  of  .7  to  four  decimal  places. 


r oo  0s-  = 

7000  X 2 = / O O O 

30  0 

/ 63 O O 

S' 3 O O x 2 = / 6 60  0 

6 o 

/ 6 6 6 0 

73  6 o X 2 = / 6720 


/ 67  2 6 


.7ooo 
3 0 0 
6 o 



. 73  6 

/ / '/  O O O 0 
Of  0 & 0 0 

/ / 070 o 

/ O O 376 

/ o o 77 


70000000 
67000000 
6000000 

777  OOOO 


EVOLUTION 


301 


934.  Rule. — Separate  into  two-figure  periods,  beginning  at  the  decimal  point, 
and  supply  ciphers  to  make  the  number  of  periods  desired. 

935.  To  extract  the  square  root  of  a common  fraction. 

Example. — What  is  the  square  root  of  J-Ji  ? 

i / 121  11 

1 TT4  — TT 

936.  Rule. — Take  the  square  root  of  the  numerator  and  of  the  denominator  ; or, 
if  this  cannot  be  done,  reduce  to  a decimal  and  extract  the  root  as  above. 


PROBLEMS  IN  SQUARE  ROOT 


937.  Extract  the  square  root  of 

1.  1 369.  5.  74926.  15.  .768. 

2.  3136.  9.  J (to  3 places).  16.  43.1. 

3.  277729.  10.  f (to  3 places).  17.  95.267. 

I 9339136.  11.  72645.  18.  43.14682. 

5.  f (to  3 places).  12.  669847.  19.  9.00746. 

6.  1429785.  13.  224.85.  20.  10.001. 

7.  984.  If  76.745. 

21.  A square  piece  of  ground  contains  12  acres.  What  is  the  distance  around 
it  in  yards  ? 

22.  A carpenter  has  9 boards,  each  16  feet  long  and  18  inches  wide.  What 
is  the  largest  square  platform  he  can  make  of  them,  allowing  nothing  for  waste? 

23.  A rectangular  field  contains  9 acres.  Its  length  is  three  times  its  width. 
Find  the  distance  around  it. 


21)..  A rectangular  field  is  16.7  rods  long,  12.47  rods  wide.  What  is  the  side 
of  a square  field  that  contains  the  same  area? 

25.  If  it  cost  $320  to  inclose  a farm  104  rods  long,  98  rods  wide,  how  much 
less  will  it  cost  to  inclose  a square  farm  of  the  same  area  with  the  same  kind  of 
fence  ? 


Similar  Figures 

938.  Similar  figures  are  those  which  have  the  same  form  and  differ  from 
each  other  only  in  magnitude.  Circles,  squares,  right-angled  triangles,  etc.,  are 
similar  figures. 

939.  Rule. — The  areas  of  all  similar  surfaces  are  to  each  other  as  the  squares  of 
their  like  dimensions. 

Like  dimensions  of  similar  surfaces  are  to  each,  other  as  the  square  roots  of  their 


areas. 


302 


EVOLUTION 


WRITTEN  PROBLEMS 

940.  1.  The  area  of  a circle,  whose  diameter  is  5 rods,  is  19.635  sq.  rd. 
What  is  the  diameter  of  a circle  whose  area  is  49  sq.  rd.? 

2.  The  area  of  a circle,  whose  diameter  is  42  feet,  is  1385.44  sq.  ft.  What 
is  the  area  of  a circle  whose  diameter  is  18  feet? 

3.  If  a pipe  2 inches  in  diameter  will  empty  a cistern  in  9 hours,  in  what 
time  will  six  pipes,  each  one-half  inch  in  diameter,  empt}?  it? 

1^.  If  a horse  tied  to  a post  by  a rope  20  feet  long  can  graze  upon  a certain 
area,  how  long  should  the  rope  be  to  allow  him  to  graze  upon  an  area  four  times 
as  great  ? 

5.  If  a pipe  6 inches  in  diameter  is  5 hours  in  running  off  a certain  quan- 
tity of  water,  in  what  time  will  3 pipes  each  3 inches  in  diameter  discharge  the 
same  quantity  ? 

6.  How  many  pipes  one-half  inch  in  diameter  are  equal  to  one  pipe  2 
inches  in  diameter? 


CUBE  ROOT 

941.  The  cube  root  of  a number  is  one  of  its  three  equal  factors.  Thus,  a 
cubic  yard  equals  3x3x3,  or  27  cubic  feet;  and  since  3 is  one  of  the  three  equal 
factors  of  27,  it  is  the  cube  root. 

The  cubes  of  2,  4,  5,  7,  9,  10,  25,  3.5,  5.4 

are  8,  64,  125,  343,  729,  1000,  15625,  42.875,  157.464. 

From  this  it  is  seen  : 

1.  That  the  cube  of  a number  of  a single  figure  will  contain  one,  two  or  three  figures. 

2.  That  the  cube  of  a number  of  two  figures  will  contain  four,  five  or  six  figures. 

3.  That  the  cube  of  any  number  will  contain  three  times  as  many  figures  as  the  number  itself, 
or  three  times  as  many  less  one  or  two. 

4-  That  if  a number  be  separated  into  three-figure  groups,  beginning  at  the  right,  or,  in  case  of 
decimals,  at  the  decimal  point,  the  number  of  figures  in  the  root  will  equal  the  number  of  periods  or 
partial  periods. 

942.  To  find  the  cube  root  of  a number. 

Example  1. — Extract  the  cube  root  of  1728. 


Explanation. — Separating  the  number  into  three-figure  groups,  it  is  seen  that  the  cube  root 
will  contain  two  figures  (tens  and  units). 


103  = 
3X  (10)2  =300 
3X10X2  = 60 
2 2 = 4 


1‘728  10 
1 000  2 
728  12  cube  root 


364 


728 


EVOLUTION 


303 


Figure  1 — t3 


The  first  or  left-hand  period  is  1,  and  the  larg- 
est cube  in  1 is  1.  This  is  represented  in  Figure  1 
by  the  cube.  Write  1 in  the  tens  place  of  the  root 
and  place  a cipher  in  the  remaining  root-place. 

This  cube,  having  10  units  (1  ten)  for  each 
dimension,  contains  (10)3  or  1000  cubic  units  (t3). 
This  subtracted  from  1728  cubic  units  leaves  728 
cubic  units,  the  contents  of  Figure  2. 


Figure  2 — 3t2Xu 

The  largest  part  of  Figure  2 is  three  square 
slabs,  each  10  units  square.  The  area  of  each  is 
10X10  (t2),  and  of  the  three  it  is  3 X (10X10)  or 
300  = 3(t)2.  Place  this  area  to  the  left  and  use  it  as 
atrial  divisor.  Dividing  300  into  728,  the  approx- 
imate thickness  of  the  slabs  is  obtained,  which  is  2 
units.  Place  2 under  the  former  root  in  units  place. 


Figure  3—3  (tXu2) 


Removing  these  three  square  slabs,  there  remain 
three  rectangular  slabs  and  a cube,  Figure  3.  Each 
slab  is  as  long  as  the  square  slabs  (10  units),  and  as 
wide  and  thick  as  the  thickness  of  the  square  slabs, 
2 units.  The  area  of  each  rectangular  slab  is  1 0 X 2, 
and  of  the  three  it  is  3 X 10X2  or  00,  (3  tXu).  Piace 
60  under  the  trial  divisor. 


Removing  the  rectangular  slabs,  the  small  cube 
remains,  Figure  4.  Each  dimension  of  this  cube  is 
the  same  as  the  thickness  of  the  slabs,  2 units.  The 
area  of  the  cube  is  2X2  or  4 (u2).  Place  4 under 
the  former  partial  divisors. 

The  sum  of  these  areas  (364)  is  the  true  divisor. 
Multiplying  this  by  the  root-figure  2,  gives  728. 
Since  there  is  no  remainder,  our  root  is  complete. 
Adding  the  partial  roots  we  have  12,  which  is  the 
cube  root  of  1728. 


Figure  4 — u3 


304 


EVOLUTION 


Example  2. — Extract  the  cube  root  of  78402.752. 

Explanation. — Commencing  at  the  decimal  point  and  separating  the  number  into  three-figure 
groups,  it  is  seen  that  the  cube  root  will  contain  three  figures  (tens,  units  and  tenths). 


40.0  3 = 

40.  Oz  X 3 = 4X00.00 
40.0X3X2.0  = 2 40.00 

2.0Z  = 4.00 

SO  44.00 


42. Oz  X 3 = 42f2.00 
42.0  X3  XX  = / O O.XO 

.Xz  = 

73.44 


yX  402 . 7sd 
04  ooo.oo  o 
/4  402 . 742 


40.0 

2.0 


.7 

42 . 7 


/ 0 0 73 . 00  0 
7 3 /4.  74 2 


4 3 /4.  74 2 


The  first  period  is  78  and  the  largest  cube  in  78  is  64,  whose  cube  root  is  4.  This  is  represented 
by  the  cube  in  Figure  1.  Write  4 in  the  tens  place  of  the  root  and  place  ciphers  in  the  remaining 
places. 

This  cube,  having  40.0  units  (4  tens)  for  each  dimension,  contains  (40.0)3  or  64000.000  cubic 
units  (t3 ).  This  subtracted  from  the  original  number  leaves  14402.752  cubic  units,  the  contents  of  the 
remainder,  a portion  like  Figure  2,  but  consisting  of  two  sets  of  slabs  of  different  thickness,  one 
within  the  other. 

The  largest  part  of  this  remaining  portion  is  made  up  of  the  three  inner  square  slabs,  seen  in 
Figure  2,  each  40.0  units  long  and  40.0  units  wide.  The  area  of  each  is  40.0  times  40.0  or  1600.00 
square  units,  and  of  the  three  it  is  3 times  1600.00  or  4800.00  (3  X tens2 ).  Place  this  area  to  the  left 
aud  use  it  as  a trial  divisor.  This  product  (4800.00)  divided  into  the  remainder  gives  the  approximate 
thickness  of  the  three  slabs  (2.0  units).  Place  2 under  the  former  root  in  units  place  and  place  a 
cipher  in  the  remaining  place. 

Removing  these  three  slabs,  a cube  and  three  rectangular  slabs  remain,  each  of  which  is  as  thick 
as  the  slabs  removed,  2 units.  Each  rectangular  slab  is  40.0  units  long  and  2 units  wide,  and  its  area 
is  40.0  times  2.0  or  80.00  square  units  ; and  the  area  of  the  three  is  3 times  80.00  or  240.00  square  units 
(3  X tensX  units).  Place  this  area  to  the  left  and  beneath  the  trial  divisor. 

Removing  these  rectangular  slabs,  there  remains  of  the  inner  part  of  Figure  2 a cube,  Figure  4, 
each  dimension  of  which  is  the  same  as  the  thickness  of  the  slabs  removed,  2.0  units.  The  area  of  this 
cube  is  2.0  times  2.0  or  4.00  square  units  (u2).  Place  this  to  the  left  and  beneath  the  other  partial 
divisors. 

The  sum  of  these  partial  divisors  is  the  area  of  all  parts  of  the  inner  portion  of  Figure  2 (the  true 
divisor).  This  sum  (5044.00)  multiplied  by  the  thickness  2.0,  gives  10088.000  cubic  units,  the  con- 
tents of  the  part  removed  This  subtracted  from  14402.752  cubic  units,  leaves  4314.752  cubic  units, 
the  contents  of  an  outer  portion  like  Figure  2. 

The  largest  part  of  this  outer  portion  is  composed  of  three  square  slabs,  each  as  long  as  the  cube, 

40.0,  plus  the  thickness  of  the  former  slab,  2.0,  or  42.0  units.  The  area  then  is  42.0  times  42.0  times  3 
or  5292.00  square  units  3 (t  -f-  u) 2.  Place  this  area  to  the  left  and  use  it  as  a trial  divisor.  This  area, 

5292.00,  divided  into  the  remainder,  4314.752,  gives  the  approximate  thickness,  .8  units.  Place  this  in 
the  proper  place  in  the  root. 

After  removing  the  square  slabs,  the  rectangular  slabs  of  Figure  3 form  the  largest  part.  Each 
of  these  is  42.0  long  and  as  thick  as  those  just  removed,  .8  units.  Hence,  the  area  of  these  is  42.0  times 
.8  times  3 or  100.80  square  units  3 ( t + u X tenths).  Place  this  area  beneath  the  trial  divisor. 


EVOLUTION 


305 


Removing  the  rectangular  slabs,  there  remains  the  small  cube,  which  is  of  the  same  dimensions 
as  the  thickness  of  the  rectangular  slabs,  .8  units.  Its  area  then  is  .8  times  .8  or  .64  square  units 
(tenths2).  Place  this  beneath  the  partial  divisors. 

The  total  area  is  5292. 00+100. 80+. 64  or  5393.44  square  units  (the  true  divisor).  This  multi- 
plied by  the  thickness,  .8,  gives  4314.752  cubic  units. 

The  cube  root  is  the  sum  of  the  partial  roots,  40.0+2.0+.8  or  42.8. 

943.  Rule. — Separate  the  number  into  three-figure  groups  counting  left  and 
right  from  units. 

Determine  the  root  of  the  largest  cube  in  the  left-hand  period  and  write  it  for  the 
first  figure  of  the  root.  Place  ciphers  in  the  remaining  root-places.  Cube  this  entire 
root,  and  subtract  from  given  number  for  a new  dividend. 

For  a trial  divisor,  square  the  root  and  multiply  by  3 ; determine  how  often  it  is 
contained  in  the  new  dividend,  and  place  this  number  under  the  root  already  found. 

To  the  trial  divisor  add  3 times  the  product  of  the  first  part  of  the  root  by  the 
second  part  of  the  root,  and  to  this  result  add  the  square  of  the  second  part  of  the  root. 
Multiply  this  complete  divisor  by  the  second  part  of  the  root  and  subtract  the  product 
from  the  dividend. 

For  the  next  trial  divisor,  square  the  sum  of  the  two  partial  roots  and  multiply 
by  three ; divide  this  product  into  the  new  dividend;  the  quotient  is  the  next  root 
figure. 

To  the  trial  divisor  add  three  times  the  product  of  the  third  part  of  the 
root  multiplied  by  the  sum  of  the  other  two  parts,  and  to  this  add  the  square  of 
the  third  part. 

Multiply  this  complete  divisor  by  the  third  part  of  the  root,  and  subtract  for  the 
new  dividend. 

If  there  are  more  root  figures  to  be  found,  for  a trial  divisor  square  the  sum  of 
the  partial  roots  and  multiply  by  3,  and  proceed  as  before. 


Note. — If  there  should  be  a remainder  after  all  root  places  are  filled,  annex  periods  of  ciphers, 
placing  one  cipher  in  the  root  for  each  period  of  3 placed  in  the  dividend. 

The  cube  root  of  a fraction  is  the  cube  root  of  the  numerator  and  of  the  denominator. 


WRITTEN  EXERCISE 

944.  Find  the  cube  root  of 


1.  91125.  5.  10218313. 

2.  24389.  6.  131096512. 

3.  274,625.  7.  187149.248. 

f.  103823.  8.  277167808. 


.9.  633.839779. 

10.  .997002999. 

11.  118805247296. 

12.  4.080659192. 


13. 

u. 

15. 

16. 


7 2 9 
3 3 7 5" 

17  2 8 
21952- 

1 ft  65  6 
1 °TT3T- 

QQ  172  8 
ou2 1952- 


306 


SIMILAR  SOLIDS 


WRITTEN  PROBLEMS 

945.  1.  A cubical  box  contains  64000  cubic  feet.  What  are  its  dimensions  ? 

2.  A cellar  24  feet  long  and  12  feet  wide  was  excavated  to  a depth  of  6 feet. 
Wt  iat  would  be  the  depth  of  a cellar  of  equal  capacity  if  it  were  cubical  ? 

3.  A cubical  tank  holds  108.4156  barrels  of  water.  What  is  the  cost  of 
lining  the  sides  and  bottom  with  tin  costing  8 cents  a square  foot? 

4-  An  open  cubical  bin  holds  3810.24  bushels  of  grain.  At  $34  per  M find 
cost  of  the  2-inch  lumber  required  to  make  the  bin. 

5.  What  is  the  entire  surface  of  a cube  whose  contents  are  91125  cubic  feet? 

6.  At  6 cents  a square  foot,  what  will  it  cost  to  plaster  the  sides  and  bottom 
of  a cubical  reservoir  which  contains  444.675  barrels  of  water? 

7.  What  is  the  length  of  a cubical  block  containing  2 cubic  feet,  1457  cubic 
inches  ? 

8.  The  length  of  a square  stick  of  timber  is  32  times  its  width  or  thickness. 
If  it  contains  13f  cubic  feet,  what  are  its  dimensions? 

9.  A bin  in  a granary  is  3 times  as  long  as  deep  and  If-  times  as  wide  as 
deep.  If  it  holds  270  bushels,  find  its  dimensions. 

10.  A cubical  pile  of  wood  contains  500  cords.  What  decimal  part  of  an 
acre  does  it  cover  ? 

Similar  Solids 

946.  Similar  solids  are  those  which  have  the  same  shape  and  differ  from 
each  other  only  in  volume. 

947.  The  volumes  of  similar  solids  are  to  each  other  as  the  cubes  of  their  like 
dimensions. 

948.  Like  dimensions  of  similar  solids  are  to  each  other  as  the  cube  roots  of 
their  volumes. 

WRITTEN  PROBLEMS 

949.  1.  If  a ball  3 inches  in  diameter  weighs  12  pounds,  how  much  will 
a ball  6 inches  in  diameter  weigh  ? 

2.  If  a haystack  15  feet  in  diameter  contains  20  tons,  how  much  hay 
should  be  in  a similar  stack  25  feet  in  diameter? 

3.  If  a cubical  bin  10  feet  long  contains  125  bushels,  how  much  grain 
should  a cubical  bin  5 feet  long  hold  ? 

4.  The  clay  in  a cubical  bank  10  feet  long  contains  50  loads.  How  long 
should  a cubical  clay  bank  be  to  contain  120  loads? 

5.  The  contents  of  a cubical  box  5 feet  long  weigh  120  pounds,  what  should 
be  the  length  of  a similar  cubical  box  to  contain  700  pounds? 


ARITHMETICAL  PROGRESSION 


950.  When  the  difference  between  any  two  consecutive  numbers  in  a series 
is  the  same,  the  numbers  form  an  arithmetical  series.  If  the  numbers  become 
greater  from  left  to  right,  it  is  an  ascending  series;  if  they  become  less,  it  is  a 

descending  series. 

951.  The  difference  between  any  two  consecutive  numbers  is  the  common 
difference. 

952.  The  quantities  considered  are  the  first  term  (a),  the  common  differ- 
ence (d),  the  number  of  terms  (n)  the  last  term  (L)  and  the  sum  of  the 
terms  (S). 

953.  To  find  the  last  term  of  an  arithmetical  progression. 

Example. — Given  a = 10,  d = 5,  n = 20,  find  L. 

Explanation. — First  term  = 10  ; second  term  = 10  + 5 ; third  term  = 10  -f  5 + 5 ; fourth 
term  = 10  -(-  5 -(-  5 -j-  5.  Then  the  20th  term  = 10  + (19  X 5)  or  105,  for  in  any  term  the  com- 
mon difference  is  used  one  time  less  than  the  number  of  that  term.  Substituting  letters  for  terms  used, 
the  following  formula  is  deduced  : 

Formula. — L = a + (n — 1)  d. 


Substituting  figures  for  the  letters  used,  L = 10  -)-  (20 — 1)  5,  or  L = 105. 


Note. — By  the  use  of  this  formula,  or  the  following  ones,  the  first  term,  or  the  common  differ- 
ence, or  the  number  of  terms  may  be  found  by  substitution.  Thus  : 

a = L — (n — 1)  d 

n —1 

L — a . , 

n---d  +i 

954.  To  find  the  sum  of  an  arithmetical  series. 

Example. — What  is  the  sum  of  seven  terms  of  the  series  5,  S,  11,  14,  etc.? 


Explanation. — Writing  the  series  as  follows  : 

Sum  = 5 -p  8 -f-  11  -f-  14  -+-  17  -(-  20  -f-  23  : in  reverse  order 

Sum  = 23  + 20  + 1?  + 11  + 11  + 8 + 5 

Adding  2S  = 28  + 28  4-  28  + 28  -f-  28  -f-  28  -f-  28  ; or,  twice  the  sum  = 28  X 7.  Now  28  is 

the  sum  of  first  term  (a)  + last  term  (L),  and  7 is  the  number  of  terms  (n)  ; hence  the  formula  : 


s = L+ax  n_ 


Substituting  figures  for  the  letters  used,  S = ^ X 7 = 98. 


307 


308 


ARITHMETICAL  PROGRESSION 


955.  R ule — The  last  term  of  an  arithmetical  series  equals  the  first  term  increased 
by  the  product  of  the  common  difference  multiplied  by  one  less  than  the  number  of  terms. 

The  sum  of  an  arithmetical  series  equals  the  sum  of  the  first  and  last  terms  multi- 
plied by  one-half  the  number  vf  terms. 

WRITTEN  PROBLEMS 

956.  1.  Given  a = 7,  d = 4,  and  n = 10 ; find  L and  S. 

2.  Given  n = 15,  a = 2J,  and  L = 44J ; find  d. 

3.  Given  L = 90f,  n = 22,  d = 4^ ; find  a and  S. 

f Given  a = 10,  L = 94,  d = 3J  ; find  n. 

5.  Find  the  8th  term  of  the  series  44,  6f,  94,  etc. 

6.  Find  the  sum  of  12  terms  of  the  series  2,  5,  8,  11,  etc. 

7.  A boy  worked  6 weeks  receiving  10  cents  the  first  day,  and  an  increase 
of  5 cents  each  day.  What  were  his  wages  the  last  day?  His  total  wages? 

8.  A man  paid  his  debt  in  15  years  by  paying  $30  the  first  year,  $38  the 
second  year,  $46  the  third  year  and  so  on ; what  was  the  debt? 

9.  A merchant  added  $250  annually  to  his  capital  for  20  years,  and  then 
had  $6000.  What  was  his  original  capital? 

10.  Fourteen  properties  were  valued  each  $225  higher  than  the  preceding 
one.  The  eighth  property  is  valued  at  $4575.  What  is  the  total  valuation? 

11.  A man  travels  15  miles  the  first  day  and  increased  the  distance  10 
miles  each  succeeding  day.  The  last  day  he  traveled  85  miles.  How  many 
days  did  he  travel?  How  far? 

12.  A clerk  receives  a salary  of  $15  the  first  month,  $25  the  second  month, 
$35  the  third  month  and  so  on  for  two  years.  What  was  his  salary  the  last 
month,  and  the  average  monthly  salary? 

13.  A man  owing  $3240  paid  it  in  a year  by  paying  $50  the  first  month, 
and  increasing  each  monthly  payment  by  a specified  sum.  What  was  the  sum? 

If  A bo}r,  who  earns  15  cents  the  first  day,  has  expenses  averaging  95  cents 
daily ; if  his  wages  increase  10  cents  daily,  what  will  be  his  total  net  earnings  in 
40  days?  On  which  day  will  he  be  most  in  debt? 

15.  Fifteen  horses  were  purchased  whose  values  are  in  arithmetical  progres- 
sion. The  fifth  horse  cost  $149,  the  twelfth  cost  $191.  What  was  the  total  cost 
of  the  horses  purchased  ? 

16.  Find  the  sum  of  20  consecutive  odd  numbers,  commencing  with  19. 

17.  In  selling  a Persian  rug  a dealer  numbers  tickets  from  1 to  200,  and 
obtains  200  persons  to  draw  each  a ticket.  Each  person  pays  for  his  ticket  as 
many  cents  as  the  number  on  the  ticket  he  draws,  and  the  owner  is  then  deter- 
mined by  lot.  How  much  does  he  receive  for  his  rug? 


GEOMETRICAL  PROGRESSION 


957.  A geometrical  series  is  one  iti  which  each  number  after  the  first  is 
derived  by  multiplying  the  preceding  by  a given  number  or  by  dividing  the 
preceding  by  a given  number. 

958.  The  multiplier  or  the  divisor  is  called  the  ratio. 

959.  When  the  numbers  increase  from  left  to  right,  the  series  is  ascending, 
as  2,  6,  18,  54 ; if  they  decrease,  it  is  descending,  as  108,  36,  12,  4. 

960.  The  quantities  considered  are  the  first  term  (a),  the  ratio  (r),  the 
number  of  terms  (n),  the  last  term  (L),  and  the  sum  of  the  terms  (S). 

961.  To  find  the  last  term  of  a geometrical  progression. 

Example. — What  is  the  tenth  term  of  the  series  2,  10,  50,  etc.? 

Explanation. — First  term=2  ; second  term=2X5  ; third  term=2X5X5  ; fourth  term  2X5X 
5X5  ; then  the  tenth  term=2X59  or  3906250,  for  in  any  term  the  ratio  is  used  one  time  less  than  the 
number  of  that  term.  Substituting  letters  for  the  terms  used,  the  following  formula  is  deduced  : 


Subtracting  (1)  from  (2)  or  3 S=32768 — 2 or  32766  and  S equals  J of  32766  or  10922. 

32766  is  the  last  term,  4 is  the  ratio,  and  3 (the  divisor)  is  one  less  than  the  ratio.  Hence  the 


L = l n_1  X a 


From  this  formula  the  following  formulas  are  deduced  : 

L 


Note. — n may  be  found  by  observing  how  many  times  the  ratio  may  be  divided  successively 


a 


962.  To  find  the  sum  of  a geometrical  series. 

Example. — What  is  the  sum  of  7 terms  of  the  series  2,  8,  32,  etc.? 


Explanation. — The  sum  (S)=2-f-8+32 -(-128+512+2048+8192 

4 (ratio)  sum  ( 4S )=  8+32+128+512+2048+8192+32768 


(1) 

(2) 


formula  : 


S=L  Xr — a 


Substituting  figures  for  the  letters  used  S= 


8192  4—2 
4—1 


or  10922 


309 


310 


GEOMETRICAL  PROGRESSION 


963.  Rule. — The  last  term  equals  the  first  term  multiplied  by  the  ratio  raised  to 
a power  one  less  than  the  number  of  terms. 

The  sum  of  a geometrical  series  equals  the  last  term  times  the  ratio  and  this 
product  divided  by  the  ratio  less  one. 

WRITTEN  PROBLEMS 

964.  1.  Given  a = 6,  r — 6,  n = 5 ; find  last  term. 

2.  Given  a — 4,  n = 7,  L = 2916;  find  the  ratio. 

3.  Given  L = 640,  n = 8,  r = 2 ; find  the  first  term. 

I/..  Given  L = 6576,  r = 6,  a = 6 ; find  the  number  of  terms. 

5.  Given  a = 8,  r = 4,  n = 4 ; find  the  last  term  and  the  sum  of  all  the 
terms. 

6.  If  4 of  the  contents  of  a vat  is  evaporated  each  day  for  8 days,  what  part 
of  the  original  contents  remains? 

7.  A milk  dealer  received  4 cent  for  the  first  pint  of  milk  sold,  and  trebled 
the  price  on  each  pint  he  sold.  What  did  he  receive  for  a gallon  sold  in  pints? 

8.  A man  having  $1000  to  pay  at  the  end  of  a year,  began  by  paying  $5 
the  first  month,  $10  the  second  month,  $20  the  third  month  and  so  on.  At  the 
end  of  the  year  has  he  overpaid  or  does  he  still  owe  and  how  much? 

9.  A debt  of  $93410  was  canceled  by  paying  $10  the  first  year,  and  treb- 
ling each  following  yearly  payment  ; how  many  years  did  it  require  to  pay  the 
debt? 

10.  The  cost  of  the  20th  cow  was  $7864.32.  If  the  first  cow  cost  14  cents 
and  the  prices  were  in  geometrical  progression,  what  was  the  ratio? 

11.  A merchant  who  began  business  with  $6000  capital  increased  the  capi- 
tal annually  by  4 of  itself.  What  was  his  capital  at  the  end  of  the  sixth  year? 

12.  A man  deposited  $1000  in  a savings  bank  that  pays  4%  compounded 
semiannually.  If  the  depositor  withdrew  no  money  from  the  bank  for  six  years, 
what  was  the  amount  he  had  on  deposit? 

13.  If  a rubber  ball,  falling  upon  a marble  floor,  rebounds  to  4 the  height 
from  which  it  has  fallen,  then  falls  again  and  rebounds  4;  etc.,  how  far  will  it 
move  before  coming  to  rest,  if  tossed  to  a height  of  50  feet? 

Note. — Last  term  = 0. 


MENSURATION 


965.  Mensuration  is  the  process  of  finding  the  lengths  of  lines,  the  areas 
of  surfaces,  and  the  volumes  of  solids. 

966.  A line  has  length  without  breadth  or  thickness. 

967.  A surface  has  length  and  breadth,  without  thickness.  A plane  surface 
or  plane  figure  is  one  that  is  flat  or  level. 

968.  A solid  has  length , breadth  and  thickness , or  depth. 

969.  Lines  are  either  straight  or  curved. 

970.  A straight  line  is  the  shortest  distance  between  two  points. 

971.  A curved  line  is  one  that  changes  its  direction  at  every  points 

972.  Parallel  lines  have  the  same  direction  and  are  equidistant. 7 

973.  A line  is  perpendicular  to  another  when  the  two  angles  thus  formed 
are  equal. 

974.  An  angle  is  the  difference  in  direction  of  two  inter- 
secting lines,  or  the  opening  between  them. 

975.  When  one  straight  line  intersects  another  so  as  to 
make  the  four  angles  so  formed  equal,  these  angles  are  called 

right  angles. 

976.  An  acute  angle  is  an  angle  that  is  less  c 

than  a right  angle ; as,  CBD. 

977.  An  obtuse  angle  is  one  that  is  greater  than 

a right  angle  ; as,  ABC.  Acute  or  obtuse  angles  may  A D 

be  called  oblique.  B 


Right-angled  Triangles 


978.  A triangle  is  a plane  surface  bounded  by  three 
straight  lines.  The  following  are  the  various  kinds  : right- 
angled,  acute-angled,  obtuse-angled  and  equi-angular ; or, 
if  named  according  to  the  sides,  equilateral,  isosceles  and 
scalene. 

979.  A right-angled  triangle  is  a triangle  which  has 
one  right  angle. 

980.  The  base  of  a triangle  is  the  line  upon  which  it 
rests;  as,  AB.  Any  side  may  become  the  base. 


c 


981.  The  altitude  of  a triangle  is  its  height  perpendicular  to  the  base  ; as, 
BC. 

982.  The  hypotenuse  of  a right-angled  triangle  is  the  side  opposite  the 
right  angle  ; as,  AC. 


311 


altitude 


312 


MENSURATION 


983.  In  every  right-angled  triangle  the  square  of 
the  hypotenuse  is  equal  to  the  sum  of  the  squares  of 
the  base  and  altitude. 

984.  The  hypotenuse  of  a right-angled  triangle  is 
equal  to  the  square  root  of  the  sum  of  the  squares  of 
the  other  two  sides. 

985.  The  base  of  a right-angled  triangle  is  equal 
to  the  square- root  of  the  difference  of  the  squares  of 
the  hypotenuse  and  altitude. 

986.  The  altitude  of  a right-angled  triangle  is 

33+42  — 52  equal  to  the  square  root  of  the  differenceof  the  squares 

9 +16  = 25  0f  f-]ie  hypotenuse  and  base. 

987.  Example.— A tree  is  broken  in  such  a manner  that  the  top  rests  on 
the  ground  40  feet  from  the  foot  of  the  tree;  the  part  remaining  standing  is  53J 
feet;  what  is  the  length  of  the  broken  part? 


402  = 1600 
53 f 2 - 2844f 

126 
1326 

66.6  = 66f  or  66| 

Explanation. — The  broken  part  of  the  tree,  the  upright  part  and  the  ground  from  the  broken 
end  to  the  foot  of  tree  form  the  three  sides  of  a right-angled  triangle,  thebaseand  altitude  being  known. 

Note. — Always  draw  the  geometrical  figure. 

988.  Rule. — To  find  the  hypotenuse,  extract  the  square  root  of  the  sum  of  the 
squares  of  the  other  two  sides. 

To  find  the  base  or  the  altitude,  extract  the  square  roo't  of  the  difference  between  the 
squares  of  the  other  two  sides. 


WRITTEN  PROBLEMS 

989.  1.  A ladder  leaning  against  a house  reaches  84  feet,  its  base  being  3S 
feet  from  the  house.  What  is  the  length  of  the  ladder? 

Two  rafters,  each  38  feet  long,  meet  at  the  ridge  of  a roof  18  feet  above 
the  attic  floor.  What  is  the  width  of  the  house? 

3.  Two  ships  start  at  the  same  point.  One  goes  due  north  120  miles,  the 
other  due  east  160  miles.  How  far  are  they  apart  ? 

i.  A tree  was  broken  56  feet  from  the  top  and  fell  so  that  the  end  struck 42 
feet  from  the  foot  of  the  tree.  Find  the  height  of  the  tree. 

5.  Find  the  distance  between  a lower  corner  at  one  end  and  the  opposite 
upper  corner  at  the  other  end  of  a room  40  feet  long,  36  feet  wide,  18  feet  high. 


MENSURATION 


313 


6.  Find  the  hypotenuse  of  a right-angled  triangle  whose  base  is  260  feet 
and  altitude  82  feet. 

7.  Find  the  hypotenuse  of  a right-angled  triangle  whose  base  is  1760  yards 
and  whose  altitude  is  880  yards. 

8.  Find  the  hypotenuse  of  a right-angled  triangle  whose  base  is  4 feet  and 
altitude  3 feet. 

9.  Find  the  diagonal  of  a square  which  is  12  inches  on  a side. 

10.  Find  the  diagonal  of  a rectangle  whose  length  is  4 rods  and  whose 
width  is  3 rods. 

11.  Find  the  hypotenuse  of  a right-angled  triangle  whose  length  is  1200 
yards  and  whose  height  is  1500  feet. 

13.  Find  the  base  of  a right-angled  triangle  whose  hypotenuse  is  360  yards 
and  whose  altitude  is  240  yards. 

13.  Find  the  altitude  of  a right-angled  triangle  whose  hypotenuse  is  500 
feet  and  whose  base  is  400  feet. 

7^-  Find  the  length  of  a rectangle  whose  width  is  10  feet  and  diagonal  12 

feet. 

15.  Find  the  side  of  a square  whose  diagonal  is  20  yards. 


Surfaces  of  Triangles 


990.  To  find  the 
area  of  a triangle. 


whose  altitude  is  49  yards  ? 

35  Xj  of  49  = 35x24|  = 857J  sq.  yd.  area. 

Example  2. — What  is  the  area  of  a triangle  whose  sides  are  respectively  30, 
32  and  40  feet  ? 

30  + 32  + 40  = 5 


51  — 30  = 21 
51  _ 32  - 19 
51  — 40  = 11 

51  X 21  X 19  X 11  = 223839 
j/223839  = 473.116+  sq.  feet. 

991.  Rule. — Multiply  the  base  by  one-half  the  altitude.  Or, 

When  the  three  sides  are  given  and  not  the  altitude,  take  half  the  sum  of  the  three 
sides,  from  it  subtract  each  side  separately , multiply  the  half  sum  and  these  remainder's 
together  and  extract  the  square  root. 


314 


MENSURATION 


WRITTEN  PROBLEMS 

992.  1.  What  is  the  area  of  a triangle  whose  base  is  36  feet  and  altitude  15 
feet  ? 

2.  What  is  the  area  of  a triangular  field  whose  altitude  is  50  yards  and 
wrhose  base  is  70  yards? 

3.  What  is  the  area  of  a triangular  field  whose  altitude  is  25  chains  and 
whose  base  is  20  chains  ? 

4-  What  is  the  area  of  a triangle  whose  sides  are  150  yards,  150  yards  and 
250  yards  ? 

5.  What  is  the  area  of  a triangle  whose  sides  are  40  rods,  50  rods  and  60 

rods  ? 


Surfaces  of  Quadrilaterals. 

993.  A parallelogram  is  a plane  surface 

bounded  by  four  straight  lines,  its  opposite  sides  being 
parallel. 

994.  A rectangle  is  a parallelogram  whose 

rectangle  angles  are  all  right  angles. 

995.  A square  is  a rectangle  whose  four  sides  are  equal  in 
length.. 

996.  To  find  the  area  of  a rectangle.  

Example. — What  is  the  area  of  a rectangular  field  whose 
length  is  20  chains  and  whose  width  is  15  chains? 

20x15  = 300  sq.  ch.  or  30  A. 

997.  Rule. — Multiply  the  length  by  the  breadth. 

7 998.  A rhomboid  is  a parallelogram,  the  adjoining 

sides  of  which  are  not  equal  to  each  other,  and  which 
contains  no  right  angle. 

rhomboid  999.  To  find  the  area  of  a rhomboid. 

1000.  Rule. — Multiply  the  base  by  the  altitude. 

1001.  A rhomb  is  an  equilateral  parallelogram  having 
oblique  angles. 

1002.  To  find  the  area  of  a rhomb. 

Example. — What  is  the  area  of  a rhouiboidal  figure  6 inches  long,  whose 
altitude  is  3 inches? 

6x3  = 18  sq.  in. 

1003.  Rule. — Multiply  the  base  by  the  altitude. 


MENSURATION 


315 


WRITTEN  PROBLEMS 

1004.  1.  Find  the  area  of  a rectangular  field  12. chains  80  links  long,  10 
chains  wide. 

2.  Find  the  area  of  a rectangular  field  100  yards  long,  87  yards  wide. 

3.  Find  the  area  of  a field  36  rods  square. 

f.  Find  the  surface  of  a blackboard  8 yards  long  and  4 feet  wide. 

5.  What  is  the  area  of  a rhomboid  whose  base  is  12  chains  and  altitude 
8 chains  ? 

6.  What  is  the  area  of  a rhomboid  whose  base  is  200  yards  and  altitude 
70  yards? 

7.  Find  the  altitude  of  a rhomboid  whose  area  is  1 acre  and  base  95  yards. 

8.  Find  the  altitude  of  a rectangle  whose  area  is  1 square  mile,  base  6 
chains. 

9.  Find  the  altitude  of  a rhomboid  of  200  acres,  whose  base  is  100  rods. 

10.  If  a road  3 miles  long  contains  12  acres,  how  wide  is  it  ? 


TEAPEZOID 


1005.  A trapezoid  is  a four-sided  plane  figure,  of  which 
two  sides  are  parallel  and  the  other  two  are  not  parallel. 

1006.  To  find  the  area  of  a trapezoid. 


Example. — The  parallel  sides  of  a trapezoid  are  respectively  20  and  26 
inches  and  the  altitude  is  18  inches ; what  is  the  area  ? 


20+26 

2 


23x18  = 414  sq.  in. 


1007.  Rule. — Multiply  J the  sum  of  the  parallel  sides  by  the  altitude. 

1008.  A trapezium  is  a four-sided  plane  figure  of 
which  no  two  sides  are  parallel. 

1009.  To  find  the  area  of  a trapezium. 

Example. — The  four  sides  of  a trapezium  are 
respectively  20,  10,  8,  24  ft. ; the  figure  is  divided  into 
two  triangles  by  a line  28  ft.  long.  What  is  the  area 
of  the  figure  ? 


Area  of  A = 70.42  sq.  ft. 
Area  of  B = 89.  sq.  ft. 

Area  of  figure  = 159.42  sq.  ft. 

1010.  Rule. — Separate  the  trapezium  into  two  triangles , find  the  area  of  each , 
and  take  the  sum. 


316 


MENSURATION 


Surfaces  of  Polygons 


PENTAGON 


HEXAGON 


1011.  A polygon  is  a figure  bounded  by  straight 
lines  or  arcs,  especially  more  than  four ; a figure 
having  many  angles. 

1012.  A regular  polygon  is  one  whose  sides 
and  angles  are  all  equal. 


1013.  To  find  the  area  of  a regular  polygon. 


Example. — The  sides  of  the  hexagonal  base  of  a statue  are  10  feet  each  and 
the  distance  from  the  center  of  the  base  to  the  middle  of  each  side  is  8.66  feet ; 
what  is  the  area  of  the  figure? 


30 

00X8.66 

P 


259.8  sq.  ft. 


1014.  Rule. — Multiply  the  distance  around  the  polygon  (perimeter)  by  J the 
distance  from  the  middle  of  one  of  the  sides  to  the  center  of  the  polygon. 


The  Circle 


1015.  A circle  is  a plane  surface  bounded  by  a curved 
line  called  its  circumference,  every  point  in  which  is  equally 
distant  from  a point  within,  called  its  center. 

1016.  The  diameter  of  a circle  is  a straight  line  passing 
through  its  center,  beginning  and  ending  in  the  circumference, 
and  dividing  the  circle  into  two  equal  parts,  called  semicircles- 

1017.  The  radius  of  a circle  is  a straight  line  from  its 
center  to  its  circumference.  It  is  J the  length  of  the  diameter. 


1018.  To  find  the  circumference  of  a circle. 


1019.  Rule. — Multiply  the  diameter  by  3. If  16. 

1020.  To  find  the  diameter  of  a circle. 

1021.  Rule. — Divide  the  circumference  by  3.1  f 16.  Or, 

Multiply  the  circumference  by  .31831. 

1022.  To  find  the  area  of  a circle. 

1023.  Ru  le. — Multiply  the  circumference  by  h the  radius.  Or, 

Square  the  diameter  and  multiply  by  .785 f.  Or, 

Square  the  circumference  and  multiply  by  .07958. 


MENSURATION 


317 


WRITTEN  PROBLEMS 

1024.  1.  Find  the  circumference  of  a circle  whose  diameter  is  3.5  yards. 

2.  Find  the  circumference  of  a circle  whose  diameter  is  12.3  meters. 

3.  What  is  the  circumference  of  a circle  whose  radius  is  28  inches? 

If..  The  circumference  of  a circle  is  10  feet;  what  is  the  diameter? 

5.  The  circumference  of  a circle  is  100  yards  ; find  the  radius. 

6.  The  diameter  of  a wheel  is  28  inches;  what  is  its  circumference? 

7.  What  is  the  circumference  of  a circle  whose  diameter  is  100  rods  ? 

8.  Find  the  area  of  a circle  whose  diameter  is  1 yard. 

9.  Find  the  area  of  a circle  whose  circumference  is  84  inches. 

10.  Find  the  area  of  a circle  whose  diameter  is  1 chain. 

11.  Find  the  area  of  a circle  whose  diameter  is  1 foot. 

12.  Find  the  distance  traversed  in  an  hour  by  a point  in  the  circumference 
of  a water-wheel  17  feet  in  diameter,  if  the  wheel  makes  1400  revolutions  in  an 
hour. 

1025.  To  find  the  circumference  or  the  diameter  of  a circle,  the  area 
being  given. 

Example. — Find  the  diameter  of  a circle  whose  area  is  200  sq.  ft.  Find 
circumference. 


254 

64  2513  2 

7854 ) 200.0000 

.07958  ) 200.00000 

157  08 

15916 

42  920 

40  840 

39  270 

39  790 

3 6500 

1 0500 

3 1416 

7958 

50840  25420 

47124  23874 


37100  15460 

31416  15916 


254.64(  15.95 

2513.20)50.13 

1 

25 

25  | 154 

1001 

1320 

125 

1001 

309  2964  Diameter  — 15.95  ft. 

10023 

| 31900 

2781 

1 

3185  1 18300 

Circumference  50.13  ft. 

| 15925 

1026.  Rule. — Divide  area  by  .785If  and  extract  the  square  root  of  the  quotient 
to  find  diameter.  Divide  area  by  .07958  and  extract  the  square  root  of  the  quotient  to 
find  circumference. 


318 


MENSURATION 


WRITTEN  PROBLEMS 

1027.  1.  The  area  of  a circle  is  10  acres ; what  is  the  diameter  ? 

2.  The  area  of  a circle  is  1 square  mile  ; what  is  the  radius  ? 

3.  The  area  of  a circle  is  100  square  feet;  what  is  the  circumference? 

j.  The  area  of  a square  inscribed  in  a circle  is  a square  rod ; what  is  the 
diameter  of  the  circle? 

5.  A square  whose  side  is  10  feet  is  inscribed  in  a circle  ; what  is  the  area 
of  the  circle  ? 

The  Ellipse 

1028.  An  ellipse  is  a plane  figure  bounded  by  a curved 
line,  such  that  the  sum  of  the  distances  from  any  point  of 
the  curve  to  two  fixed  points  (the  foci ) is  constant. 

1029.  The  transverse  axis  of  an  ellipse  is  the  diameter 
drawn  through  the  foci. 

1030.  The  conjugate  axis  is  the  diameter  at  right 
angles  to  the  transverse  axis. 

1031.  To  find  the  area  of  an  ellipse. 

1032.  Rule. — Multiply  J of  the  transverse  axis  by  J of  the  f 
conjugate  axis,  and  the  resulting  product  by  3. If  16. 


f,  g.  foci  ; pf+pg,  con- 
stant 


Solids 


TRIANGULAR  PRISM 


SQUARE  PRISM 


PENTAGONAL  PRISM 


1033.  The  principal  regular  solids  are  the  prism,  cube,  pyramid,  cone  and 
sphere. 

1034.  A Prism  is  a solid  whose  bases  or  ends  are  any  similar  equal  and 
parallel  plane  figures,  and  whose  sides  are  parallelograms. 

1035.  Prisms  are  triangular,  square,  pentagonal,  etc.,  according  as  the  figures 
of  their  ends  are  triangles,  squares,  pentagons,  etc. 


MENSURATION 


319 


PARALLELEPIPED 


1036.  A pri  sm  whose  bases  are  parallel- 
ograms is  called  a parallelepiped. 

1037.  A cube  is  a solid  bounded  by  six 
equal  squares. 

1038.  To  find  the  entire  surface  of  a 

prism.  cube 

1039.  Rule. — Multiply  the  distance  around  the  base  by  the  altitude,  and  to  the 
product  add  the  areas  of  the  top  and  bottom. 


~7 

/ 

y 

/ 

1040.  To  find  the  volume  or  contents  of  any  prism. 

1041.  Rule. — Multiply  the  area  of  the  base  by  the  altitude  of  the  prism. 

1042.  To  find  the  volume  or  contents  of  a parallelepiped. 

1043.  Rule. — Multiply  together  the  length,  breadth  and  height. 


CYLINDER 


1044.  A cylinder  is  a round  prism  of  uniform  diameter 
with  circles  for  its  ends. 

1045.  The  altitude  of  a cylinder  is  the  distance  from  the 
center  of  one  base  to  the  center  of  the  other. 

1046.  To  find  the  entire  surface  of  a cylinder. 

1047.  Rule. — Multiply  the  circumference  of  the  base  by  the 
altitude  and  to  the  product  add  the  areas  of  the  tivo  ends. 


1048.  To  find  the  volume  or  contents  of  a cylinder. 

1049.  Rule. — Multiply  the  area  of  the  base  by  the  altitude. 


WRITTEN  PROBLEMS 

1050.  1.  Find  the  entire  surface  of  a parallelepiped  whose  length  is  6 feet 
and  sides  of  the  base  4 feet. 

Find  the  volume  of  a granite  block  6 feet  high,  2 feet  wide,  1 foot  thick. 

3.  Find  the  volume  of  a triangular  prism  whose  area  is  3 square  feet  and 
wdiose  length  is  10  feet. 

f Find  the  entire  surface  of  a cylinder  18  inches  in  diameter,  6 feet  long. 

5.  Find  the  volume  of  a cylinder  4 feet  long  whose  circumference  is  2 feet. 

1051.  A pyramid  is  a solid  bounded  by  a plane 
polygon  for  its  base,  and  by  triangular  planes  meeting  in 
a point  called  the  vertex. 

1052.  To  find  the  entire  surface  of  a pyramid. 

1053.  Rule. — Multiply  the  perimeter  of  the  base  by  1 
the  slant  height,  and  to  the  product  add  the  area  of  the  base. 

1054.  To  find  the  volume  or  contents  of  a 
pyramid. 


PYRAMID 


320 


MENSURATION 


1055.  Rule. — Multiply  the  area  of  the  base  by  £ of  the  altitude. 

1056.  A cone  is  a solid  that  tapers  uniformly  from  a 
circular  base  to  a point  called  the  vertex. 

1057.  To  find  the  entire  surface  of  a cone. 

1058.  Rule. — Multiply  the  circumference  of  the  base  by 
| the  slant  height , and  to  the  product  add  the  area  of  the  base. 

1059.  To  find  the  volume  or  contents  of  a cone. 

1060.  Rule. — Multiply  the  area  of  the  base  by  J of  the 
altitude. 

1061.  A sphere  is  a solid,  every  point  of  whose  sur- 
face is  equally  distant  from  an  interior  point,  called  the 
center  of  the  sphere. 

1062.  The  diameter,  or  axis,  of  a sphere  is  a 
straight  line  passing  through  the  center  and  terminated 
both  ways  by  the  surface. 

1063.  The  radius  of  a sphere  is  a straight  line 
passing  from  the  center  to  any  point  on  the  surface.  Its 
length  is  \ of  the  diameter. 

1064.  To  find  the  surface  of  a sphere. 

1065.  Rule. — Multiply  the  circumference  by  the  diamt 


CONE 


Multiply  the  square  of  the  diameter  by  3.14-16. 

1066.  To  find  the  volume  or  contents  of  a sphere. 

1067.  Rule. — Multiply  the  surface  by  } of  the  diameter.  Or, 


Multiply  the  cube  of  the  diameter  by  .5236. 


SPHEROID 


1068.  A spheroid  is  a solid  formed  by  the  revolu- 
tion of  an  ellipse  about  one  of  its  axes. 

1069.  To  find  the  volume  or  contents  of  a 
spheroid. 

1070.  Rule. — Multiply  the  square  of  the  revolving  axis 
by  the  fixed  axis , and  the  resulting  product  by  .5236. 


WRITTEN  PROBLEMS 

1071.  l.  Find  the  whole  surface  of  a pyramid,  a side  of  the  square  base 
being  6 feet  and  the  slant  height  8 feet. 

2.  Find  the  whole  surface  of  a pyramid  having  a rectangular  base  5 feet 
by  4 feet  and  a slant  height  of  4 feet  3 inches. 


MENSURATION 


321 


3.  Find  the  volume  of  an  octagonal  pyramid,  the  area  of  the  base  being  2 
square  feet  and  the  altitude  being  2 feet. 

If.  Find  the  entire  surface  of  a cone,  the  radius  of  the  base  being  2 feet  and 
the  slant  height  5 feet  6 inches. 

5.  Find  the  volume  of  a cone  whose  diameter  is  12  inches  and  altitude 
2 feet. 

6.  Find  the  surface  of  a sphere  3 feet  in  diameter. 

7.  Find  the  volume  of  a sphere  whose  circumference  is  3.1416  feet. 

REVIEW  PROBLEMS  IN  MENSURATION 

1072.  1.  What  is  the  area  of  a triangle  whose  base  is  16  feet,  altitude  36 
feet  ? 

2.  What  is  the  area  of  an  isosceles  triangle,  the  equal  sides  of  which  are 
170  yards  and  the  base  308  yards  ? 

3.  What  is  the  area  of  a triangle,  the  sides  of  which  are  205  feet,  228  feet 
and  281  feet  ? 

If.  If  the  base  of  a right-angled  triangle  is  42  feet  and  its  altitude  48  feet, 
what  is  the  hypotenuse? 

5.  A rectangular  field  is  296  feet  long,  272  feet  wide.  Find  the  distance 
between  the  opposite  corners. 

6.  A ladder  placed  against  a wall  reaches  to  a height  of  68  feet.  The 
bottom  of  the  ladder  is  22  feet  from  the  foot  of  the  wall.  Find  the  length  of  the 
ladder. 

7.  The  sides  of  a trapezium  taken  in  order  are  12,  12,  9 and  7,  and  the 
diagonal  which  cuts  off  the  first  two  sides  is  14  rods.  What  is  the  area  of  the 
trapezium  ? 

8.  The  circumference  of  a circle  is  one-half  of  a mile.  Find  its  diameter. 

9.  The  diameter  of  a circle  is  290  feet.  Find  its  circumference. 

10.  The  diagonal  of  a square  is  1000  feet.  Find  its  area  in  square  feet. 

11.  What  is  the  area  of  a piece  of  ground  in  the  form  of  a circle  225  feet  in 
diameter? 

12.  Of  two  concentric  circles,  the  diameter  of  the  inner  one  is  25  rods,  and  of 
the  outer  one,  28  rods.  Find  the  area  of  the  space  included  between  the  two 
circumferences. 

13.  What  is  the  total  surface  of  a cylinder  whose  diameter  is  12  feet  and 
whose  length  is  28  feet  ? 

Ilf.  How  many  gallons  of  water  will  a tank  hold  that  is  8 feet  square  at  the 
base  and  9 feet  deep? 

15.  A room  is  24  feet  long,  15  feet  high,  18  feet  wide.  Find  the  diagonal 
of  the  room. 


322 


MENSURATION 


16.  At  22  cents  a square  yard,  what  will  it  cost  to  gild  a ball  9 feet  in 
diameter  ? 

17.  What  is  the  weight  of  a ball  of  iron  7 inches  in  diameter,  if  iron  weighs 
450  pounds  to  the  cubic  foot? 

18.  A wagon  7 feet  long,  4 feet  6 inches  wide,  is  built  to  hold  3 tons  of  coal. 
Find  its  depth,  there  being  54  pounds  to  a cubic  foot  of  coal. 

19.  A mow  28  feet  long,  24  feet  wide,  and  12  feet  high  is  filled  with  timothy 
hay.  If  the  weight  of  a cubic  foot  is  7 pounds,  what  is  the  value  of  the  hay  in 
this  mow  at  $15  per  ton  of  2000  pounds? 

20.  A bin  12  feet  long,  7 feet  wide,  is  built  to  hold  500  bushels  of  oats.  Find 
its  depth. 

21.  What  is  the  weight  of  a conical  piece  of  metal  22  inches  in  diameter,  15 
inches  in  slant,  if  the  metal  of  which  it  is  composed  weighs  420  pounds  to  the 
cubic  foot  ? 

22.  A field  in  the  form  of  a rectangle  contains  3 acres  of  ground.  Its  length 
is  three  times  its  width.  What  will  it  cost  at  48  cents  a running  yard  to  put  a 
fence  around  it  ? 

23.  Which  costs  the  more  and  how  much,  a fence  around  a field  containing 
10  acres  whose  length  is  twice  its  width,  at  62  cents  a running  yard,  or  the  same 
kind  of  fence  around  a square  field  containing  the  same  area? 

2 If..  A chandelier  is  suspended  from  the  center  of  the  ceiling  of  a room- 
The  room  is  15  feet  high,  24  feet  long,  22  feet  wide.  If  the  bottom  of  the  chan- 
delier is  9 feet  from  the  floor,  what  is  the  distance  from  a lower  corner  of  the 
room  to  the  central  point  of  the  bottom  of  the  chandelier  ? 

25.  A reservoir  in  the  form  of  a circle  is  9 feet  deep  and  48  feet  in  diameter. 
When  full,  how  long  will  the  water  contained  in  it  last  a towm  of  800  people,  if 
to  each  inhabitant  there  are  allowed  50  gallons  every  24  hours? 

26.  The  rain  which  falls  on  a roof  90  feet  long  and  85  feet  wide,  in  4S  hours, 
when  the  rainfall  averages  f of  an  inch  every  two  hours,  will  exactly  fill  a cylin- 
drical tank  22  feet  in  diameter.  Find  the  depth  of  the  tank. 

27.  A ball  made  of  iron,  which  weighs  450  pounds  to  the  cubic  foot,  has  a 
diameter  of  9 inches.  How  many  balls  whose  diameters  are  one  inch  would  be 
required  to  weigh  as  much  ? 

28.  A field  is  126  chains  long  and  105  chains  wide.  What  is  it  worth  at 
$490  an  acre  ? 

29.  The  base  of  a triangle  is  90  feet.  Its  area  is  2242  square  feet.  Find  its 
altitude. 

30.  A rectangular  field  contains  9 acres  of  ground.  One  of  its  dimensions  is 
42  chains.  Find  the  other  dimension  in  rods. 

31.  The  dimensions  of  a dumb-bell  are  as  follows : the  handle  is  5 inches 
long  and  1 inch  thick  ; the  ends  are  spheres  3 inches  in  diameter.  If  the  dumb- 
bell is  made  of  iron  weighing  450  pounds  to  the  cubic  foot,  what  is  its  weight '? 


MENSURATION 


323 


32.  What  will  it  cost,  at  92  cents  per  square  yard,  to  macadamize  a section 
of  street  2J  miles  long,  80  feet  wide? 

33.  If  the  diameter  of  a circle  is  320  rods,  find  its  circumference. 

34..  If  the  circumference  of  a circle  is  a quarter  of  a mile,  find  its  area. 

35.  If  a pipe  3 inches  in  diameter  will  supply  a certain  amount  of  water  in 
a given  time,  how  many  pipes  one-half  inch  in  diameter  will  be  required  to 
supply  the  same  amount  of  water  in  the  same  time? 

36.  At  8£  cents  a pound,  what  is  the  cost  of  a grindstone,  the  diameter  of 
which  is  18  inches  and  thickness  4 inches,  if  in  the  center  there  is  a hole  one 
inch  square?  A cubic  foot  of  this  kind  of  stone  weighs  185  pounds. 

37.  A cabinet  maker  has  a piece  of  mahogany  18  feet  long,  7 feet  6 inches 
wide.  How  large  a square  table-top  could  be  made  from  this,  no  allowance  for 
waste? 

38.  A platform,  circular  in  form,  holds  850  people.  Allowing  2 square  feet 
for  each  person,  what  is  the  diameter  of  the  platform? 

39.  A right-angled  triangle,  the  base  being  5 feet  and  altitude  4 feet,  is 
made  to  turn  round  on  the  longer  side;  find  the  volume  of  the  cone  thus  gen- 
erated. 

40.  The  surface  of  a certain  solid  is  3 times  as  great  as  the  surface  of  a sim- 
ilar solid;  what  is  the  ratio  of  their  volumes? 

41.  A rectangular  field  is  300  rods  long  and  200  yards  wide;  what  is  the 
length  of  its  diagonal? 

42.  A lot  is  1 rod  square;  how  many  cubic  yards  of  earth  will  it  require  to 
raise  the  surface  of  the  lot  2 feet  ? 

43.  Find  the  weight  of  a spherical  iron  ball  whose  diameter  is  8 inches,  a 
cubic  foot  of  iron  weighing  450  pounds. 

44 • If  a sphere  4 inches  in  diameter  weighs  4 pounds,  what  will  be  the 
weight  of  a sphere  of  the  same  material  1 foot  in  diameter? 

45.  Find  the  weight  of  a spherical  shell  1 inch  thick,  outside  diameter  1 
foot,  the  metal  weighing  450  pounds  to  the  cubic  foot. 


LATITUDE  AND  LONGITUDE 

1073.  Latitude,  the  distance  north  or  south  of  the  equator,  is  measured 
in  degrees,  minutes  and  seconds ; no  latitude  can  exceed  90  degrees,  because 
from  the  equator  to  either  pole  is  one-fourth  of  360°  or  90°. 

1074.  Places  north  of  the  equator  are  in  north  latitude ; those  south,  in  south 
latitude.  Places  on  the  equator  have  no  latitude. 

1075.  Longitude,  the  distance  east  or  west  from  a selected  or  prime  meri- 
dian (usually  Greenwich),  is  measured  in  degrees,  minutes  and  seconds;  no  lon- 
gitude can  be  greater  than  180  degrees,  because  from  a given  meridian  to  that 
meridian  is  360°  ; then  one-half  of  360°  is  east  and  one-half  is  west. 


Note. — la  the  United  States  the  meridian  of  Washington  is  sometimes  selected,  and  maps  show 
longitude  from  Greenwich  at  the  top  and  from  Washington  at  the  bottom. 

1076.  Places  east  of  the  selected  meridian  are  in  east  longitude-,  those  west, 
in  west  longitude. 

1077.  To  find  the  difference  in  latitude  or  longitude. 

Examples. — (a)  The  latitude  of  Philadelphia  is  39°  56'  39"  N.,  and  of  Mont- 
real 45°  24'  28"  N. ; what  is  the  difference  in  latitude  ? (b)  The  longitude  of 
Berlin  is  13°  24'  28"  E.,  and  of  New  York  is  74°  3'  W. ; what  is  the  difference  in 
longitude?  45°  24'  28"  N. 

Explanation. — (a)  Since  Montreal  and  Philadelphia  are  both  north  of 
the  equator,  their  difference  of  latitude  must  be  the  distance  that  Montreal  is 
farther  north  than  Philadelphia  ; this  distance  is  found  by  subtraction. 

13°  24'  28"  E. 

74  3 00  W. 


39  56  39  N. 


87°  27'  28" 


5°  27'  49" 

(b)  From  the  meridian  of  Berlin  to  the  prime  meridian  is  13°  24'  28", 
and  from  the  prime  meridian  to  the  meridian  of  New  York  is  74°  3'  ; the  dis- 
tance from  Berlin  to  New  York,  or  the  difference  of  longitude,  is  found  by  addi- 
tion. 


1078-  Rule. — The  difference  of  latitude  or  longitude  is  found  by  subtraction 
when,  both  places  are  in  like  latitudes  or  longitudes  ; by  addition  when  the  places  are 
in  different  latitudes  or  longitudes. 


WRITTEN  PROBLEMS 

1079.  1.  The  latitude  of  New  York  is  40°  24'  40"  N.,  of  Cape  Horn  55°  5S' 
30"  S.,  and  of  Washington  38°  53'  39"  N.  What  is  the  difference  in  latitude 
between  New  York  and  Cape  Horn?  between  Washington  and  Cape  Horn? 
between  New  York  and  Washington? 

2.  The  longitude  of  Detroit  is  82°  58'  W.,  of  San  Francisco  122°  26'  15" 
W.,  and  of  Rome  12°  27'  E.  What  is  the  difference  in  longitude  between 
Detroit  and  San  Francisco?  between  Detroit  and  Rome?  between  San  Francisco 
and  Rome? 


324 


LONGITUDE  AND  TIME 


325 


3.  The  latitude  of  Philadelphia  is  39°  56'  38"  N.,  of  Callao  11°  45'  2t"  S., 
and  of  Boston  42°  21'  27"  N.  What  is  the  difference  in  latitude  between  Phila- 
delphia and  Callao?  between  Boston  and  Callao?  between  Philadelphia  and 
Boston  ? 

The  longitude  of  Philadelphia  is  75°  9'  5"  W.,  of  New  Orleans  90°  W., 
and  of  Calcutta  88°  19'  2"  E.  What  is  the  difference  in  longitude  between  Phil- 
adelphia and  Calcutta?  between  New  Orleans  and  Calcutta?  between  Philadel- 
phia and  New  Orleans? 

5.  The  longitude  of  Canton  is  113°  15'  33"  E.,  of  St.  Paul  93°  4'  55"  W. 
of  Omaha  95°  55'  15"  W.  What  is  the  difference  in  longitude  between  Canton 
and  St.  Paul?  between  Canton  and  Omaha? 

Note. — If  the  difference  in  longitude  exceeds  180  degrees,  subtract  this  difference  from  360 
degrees  to  find  the  real  difference  in  longitude. 


LONGITUDE  AND  TIME 

1080.  The  rotation  of  the  earth  on  its  axis  from  west  to  east  ^3b0  m 24 
hours)  causes  midday  or  noon,  or  sunrise,  or  any  given  time,  to  pass  westward  15° 
in  one  hour;  15'  in  1 minute;  15"  in  one  second. 

Note. — When  it  is  sunrise,  6 A.  M.,  in  Philadelphia,  places  west  of  Philadelphia,  as  Chicago, 
will  not  have  seen  the  sunrise,  hence  the  hour  will  be  before  sunrise,  or  earlier.  When  Chicago  sees  the 
sunrise,  Philadelphia  will  see  the  sun  above  the  horizon,  and  the  time  indicated  hy  clocks  will  be  past 
6 or  later  than  Chicago.  From  this  we  conclude  that  a place  east  of  a given  place  has  later  time  than 
the  given  place  ; places  west,  earlier  time.  Places  on  the  same  meridian  have  the  same  time. 


1081.  Solar  time  is  the  sun’s  time  as  indicated  by  a sun  dial. 

1082.  Standard  time  is  time  established  by  the  leading  railroad  companies 
of  the  United  States.  For  convenience  they  have  divided  the  United  States  into 
four  time-belts  each  fifteen  degrees  wide,  and  have  adopted  the  time  of  a single 
meridian  in  that  belt  as  the  standard  time  for  the  entire  belt. 


These  belts  are : 

1.  Eastern,  whose  time  is  the  local  or  solar  time  of  the  75th 
which  is  9'  east  of  Philadelphia. 

2.  Central,  whose  time  is  the  local  or  solar  time  of  the  90th 
which  is  2°  15'  west  of  Chicago. 

3.  Mountain,  whose  time  is  the  local  or  solar  time  of  the  105th 
which  is  44'  west  of  Denver. 

It-.  Pacific,  whose  time  is  the  local  or  solar  time  of  the  120th 
which  is  2°  26'  east  of  San  Francisco. 


meridian, 

meridian, 

meridian, 

meridian, 


Note  — An  outline  map  showing  time-belts  and  official  time  may  be  secured  from  a railroad 
company. 


. 326 


LONGITUDE  AND  TIME 


Dials  Exhibiting  Comparison  of  Solar  and  Standard  Time 


1083.  To  find  the  difference  in  time  between  places  differing  in  longu 
tude. 


Example. — (a)  When  it  is  11  A.  M.,  solar  time,  at  Philadelphia,  75°  9'  5" 
W.,  what  time  is  it  at  Denver,  104°  44'  20"  W.  ? ( b ) What  is  the  standard  time 
at  Denver  when  it  is  11  A.  M.  standard  time  at  Philadelphia  ? 


104° 

75 


44' 

9 


20" 

5 


W 

W 


Explanation.  — (a)  The  difference  in  longitude  between  the  two 
places  is  29°  35'  15".  Divide  this  difference  by  15,  since  every  15°  = one 
hr. ; 15'  = one  min. ; 15//  = one  sec.  This  gives  1 hr.  58  min.  21  sec.,  the 
difference  in  time  between  the  two  places.  Denver  being  west  of  Phila- 
delphia has  earlier  time  ; subtracting  1 hr.  58  min.  21  see.  from  11  A.  M. 
leaves  9 hr.  1 min.  39  sec. ; hence  the  time  at  Denver  is  1 min.  39  sec.  after 
9 A.  M. 

( b ) The  standard  time  at  Denver  (mountain  time)  is  2 hr.  earlier  than 
Philadelphia  (eastern  time),  hence  subtract  2 hr.  from  11  A.  M.  9 A.  M. 
is  the  standard  time  at  Denver. 

1084.  Rule  1. — Divide  the  num,>er  of  degrees,  minutes  and 
in  longitude  between  two  places ) by  15  ; the  quotient  will  be  the  difference  of  time  in 
1 lours,  minutes  and  seconds. 


15)  29 

' 35 

15 

1 

58 

21 

hrs. 

min. 

sec. 

11 

00 

00 

1 

58 

21 

9 

1 

39 

seconds  ( the 

difference 

Rule  2. — Multiply  the  hours , minutes  and  seconds  (the  difference  in  time 
between  two  places ) by  15  ; the  product  will  be  the  difference  in  degrees,  minutes  and 
seconds  between  two  places. 


LONGITUDE  AND  TIME 


327 


WRITTEN  PROBLEMS 

1085.  1.  The  longitude  of  Pekin  is  118°,  and  that  of  Berlin  13°  24'  28"  E. 
What  is  the  difference  in  solar  time?  When  it  is  3 P.  M.  at  Berlin,  what  time 
is  it  at  Pekin  ? 

2.  The  longitude  of  Paris  is  2°  20'  17"  E.,  and  that  of  Philadelphia  75° 
9'  5"  W.  What  is  the  difference  in  time  ? When  it  is  half-past  two  o’clock 
P.  M.  in  Philadelphia,  what  time  is  it  in  Paris?  in  Berlin? 

3.  What  is  the  standard  time  in  New  York  when  the  standard  time  in 
Portland,  Oregon,  is  25  min.  18  sec.  past  3 o’clock  P.  M.  ? 

J.  The  longitude  of  Washington  is  77°  3'  W.  If  the  difference  in  time 
between  Washington  and  Edinburg  is  4 hr.  53  min.  28  sec.,  what  is  the  longitude 
of  Edinburg?  When  it  is  20  min.  past  3 o’clock  P.  M.  at  Edinburg,  what  time 
is  it  in  Washington  ? 

5.  The  captain  of  a ship  observed  that  when  the  sun  crossed  the  meridian, 
the  solar  time,  shown  by  his  chronometer  set  to  Greenwich  time,  was  22  min.  4 
sec.  after  3 o’clock  P.  M.  In  what  longitude  was  the  ship  ? 

6.  St.  Paul,  whose  longitude  is  93°  4'  55"  W.,  is  situated  in  the  Central 
belt.  What  is  the  difference  between  the  local  or  sun  time  and  the  standard  time 
at  St.  Paul? 

7.  In  traveling  from  Philadelphia  to  Seattle,  Washington,  my  watch,  which 
shows  Philadelphia  time,  must  be  changed  on  arriving  at  Seattle.  How  much 
must  I change  it  and  which  way  (standard  time)? 

8.  The  longitude  of  Manila  is  121°  15"  E.,  and  of  San  Juan  66°  5'  W.; 
what  is  the  difference  in  longitude?  When  it  is  5 P.  M.  Tuesday  at  San  Juan, 
what  is  the  time  at  Manila? 

9.  The  longitude  of  Berlin  is  13°  24'  28"  E.,  and  of  Philadelphia  75°  9'  5" 
W.  A cablegram  sent  from  Berlin  at  12.30  noon  requires  30  minutes  in  transit. 
At  what  hour  will  it  be  received  in  Philadelphia? 

10.  Sydney  is  in  longitude  151°  38'  42"  E.,  and  San  Francisco  is  122°  26' 
15"  W.;  when  it  is  4 P.  M.  July  4th  at  San  Francisco,  what  time  is  it  at  Sydney? 

11.  A and  B met,  and  on  comparing  watches  found  that  A’s  indicated  10.30 
A.  M.  and  B’s  showed  5.10  P.  M.  The  longitude  of  A’s  place  was  82°  58' ; what 
was  the  longitude  of  B’s  place  ? 

12.  Two  persons  are  so  situated  that  they  are  14°  42'  48"  apart,  yet  are  in  the 
same  time  belt.  The  first  is  6°  14'  27"  East  of  the  standard  meridian,  and  the 
other  is  8°  28'  21"  West.  Will  the  local  time  of  each  be  too  fast  or  too  slow, 
and  how  much  ? 


REVIEW  MENTAL  PROBLEMS 

1086.  1-  A man’s  age  increased  by  J of  his  age  equals  40  years ; what  is 
his  age? 

2.  What  number  is  that  which  if  its  \ be  taken  away,  the  remainder  will 
be  33? 

3.  Three  times  a number  increased  by  its  £ equals  32  ; what  is  the  number? 

4.  What  number  is  that  which  being  increased  by  its  \ and  ^4-  equals 
27? 

5.  A man  having  spent  \ of  his  money  had  45  cents  remaining ; what 
sum  had  he  at  first? 

6.  Margaret  spent  J of  her  money  in  one  store  and  4 of  her  money  in 
another  store  and  had  21  cents  remaining;  how  much  had  she  at  first? 

7.  One-half  of  the  length  of  a pole  is  in  the  air,  4 in  the  water  and  18  feet 
in  the  ground;  required  the  length  of  the  pole. 

8.  Two-thirds  of  Adam’s  money  diminished  by  2 dollars  equals  J of  his 
money  ; what  was  his  sum  of  money? 

9.  What  number  is  that  which  diminished  by  its  4 and  increased  by  its 
4 will  equal  50  ? 

10.  What  is  Benjamin’s  age,  if  f of  his  age  increased  by  6 years  equals  36 
years  ? 

11.  The  cost  of  a horse  diminished  by  \ of  its  cost  and  five  dollars  equals 
seventy  dollars;  what  did  the  horse  cost? 

12.  Abram’s  money  increased  by  twenty  dollars  equals  f of  his  mone}T ; what 
is  his  money  ? 

13.  Bella’s  age  increased  by  32  years  equals  3 times  her  age ; what  is 
her  age  ? 

14-  Four  times  a number  diminished  by  once  the  number  equals  75  ; what 
is  the  number  ? 

15.  If  the  height  of  a tree  be  increased  by  its  f and  20  feet,  the  sum  will 
be  twice  the  height ; what  is  the  height  of  the  tree? 

16.  The  cost  of  a horse  diminished  by  £ of  its  cost  and  §30  equals  4 of  its 
cost ; wbat  did  the  horse  cost  ? 

17.  I sold  J of  my  farm  and  then  bought  30  acres  and  now  have  SO  acres ; 
how  much  land  had  I at  first  ? 

18.  If  the  height  of  a monument  be  increased  by  its  4 and  that  sum  dimin- 
ished by  its  4,  the  height  will  equal  75  feet;  wdiat  is  its  height? 

19.  A father  and  son  earned  in  one  week  $24 ; how  much  did  each  earn  if 
the  father  earned  three  times  as  much  as  his  son? 

20.  A pole  60  feet  in  length  was  sawed  into  two  unequal  parts  so  that  4 of 
the  longer  piece  equals  the  shorter ; what  was  the  length  of  each  piece  ? 


328 


REVIEW  MENTAL  PROBLEMS 


329 


21.  X and  Y had  equal  sums  of  money.  X lost  $10,  Y earned  $20  and  they 
then  together  had  $110  ; how  much  had  each  at  first? 

22.  A cow  and  a horse  cost  $120.  If  the  cow  cost  f as  much  as  the  horse, 
minus  $10,  required  the  cost  of  each. 

23.  A wheelman  rode  130  miles  in  3 days.  He  rode  10  miles  farther  the 
second  day  than  the  first  and  10  miles  less  the  third  day  than  the  second  ; how 
far  did  he  ride  each  day? 

2 Ip.  M and  N together  have  $40  ; how  many  dollars  has  each,  if  three  times 
M’s  money  equals  N’s  ? 

25.  P and  Q can  build  f of  a boat  in  a day  and  Q does  | as  much  as  P ; 
what  part  can  each  complete  in  a day  ? 

26.  Divide  $66  between  tw7o  persons  so  that  as  often  as  the  first  has  $2,  the 
second  shall  have  $3J. 

27.  A’s  money  added  to  4 of  B’s  money  equals  $70  ; how  much  money  has 
each  if  B’s  money  is  to  A’s  as  3 to  2 ? 

28.  If  § of  a yard  of  cloth  cost  f of  a dollar,  what  will  J of  a yard  cost? 

29.  If  5 men  earn  $40  in  a certain  time,  how  much  will  4 men  earn  in  twdce 
the  time? 

30.  Two  men  hire  a pasture  for  $5.  One  turns  in  3 horses  for  7 days  and 
the  other  7 horses  for  2 days;  how  much  should  each  pay  ? 

31.  Two  men  had  equal  sums  of  money;  one  lost  $7  and  the  other  lost  $4 
and  they  then  together  had  $19.  How  much  did  each  have  at  first? 

32.  Jones  and  Black  can  build  4 of  a boat  in  a day,  and  Black  builds  § as 
much  as  Jones;  how  much  can  each  build  in  a day  ? 

33.  If  5 men  can  build  40  rods  of  wrall  in  16  days,  how  many  men  can  build 
half  as  much  wall  in  4 days  ? 

3Ip.  If  a five-cent  loaf  of  bread  wreighs  8 ounces  when  flour  is  worth  $4  a 
barrel,  how  much  should  it  weigh  when  flour  is  worth  $5  a barrel  ? 

35.  Three  men  hired  a horse  for  30  days  at  the  rate  of  $2  a day.  The  first 

used  it  8 days,  the  second  10  days  and  the  third  12  days ; how7  much  should 
each  pay  ? 

36.  A and  Z contract  to  do  a piece  of  work  in  90  days ; A sends  9 men  and 

Z 15  boys.  What  part  of  the  price  should  each  receive,  supposing  5 boys  to  do 

as  much  as  3 men  ? 

37.  X,  Y and  Z dig  a ditch  for  $72  ; X gets  $1  a day,  Y $1J  a day  and  Z 
$lf  a day.  How  many  days  were  they  employed  and  what  did  each  receive? 

38.  Two  men  can  mow  a field  of  grass  in  4 hours  ; the  first  can  mow  it  alone 
in  9 hours;  how  long  would  it  require  the  second  alone  to  mow  it? 

39.  Brown  can  dig  a ditch  in  6 days  and  Black  can  dig  it  in  8 days  ; in  what 
time  can  they  dig  it  working  together? 


330 


REVIEW  MENTAL  PROBLEMS 


Ifi.  A cistern  can  be  filled  by  an  inlet  pipe  in  5 hours  and  emptied  by  an 
outlet  pipe  in  4 hours  ; if  the  cistern  is  full  and  both  pipes  are  opened,  in  what 
time  will  the  tank  be  emptied? 

41.  A cistern  can  be  filled  by  one  pipe  alone  in  6 hours  and  by  a second 
alone  in  9 hours;  how  long  long  will  it  require  both  to  fill  it? 

42.  A cistern  can  be  filled  by  an  inlet  pipe  in  4 hours  and  emptied  by  an 
outlet  pipe  in  5 hours;  if  the  cistern  is  empty  and  both  pipes  are  opened,  bow 
long  will  be  required  to  fill  it? 

43.  Brown,  Baker  and  Broad  lunch  together,  Brown  having  3 loaves,  Baker 
5 loaves  and  Broad  24  cents  to  divide  between  them.  If  the  bread  be  divided 
equally  among  them,  how  should  Brown  and  Baker  divide  the  money? 

44-  What  is  the  time  of  day,  if  the  time  since  noon  equals  ^ of  the  time 
since  midnight? 

45.  At  what  time  after  3 o’clock  are  the  hour  and  minute  hands  together? 

46.  What  is  the  time  of  day,  if  the  time  till  midnight  equals  -f  of  the  time 
since  midnight? 

47.  At  what  time  after  3 o’clock  are  the  hour  and  minute  hands  opposite? 

48.  A pole  whose  length  was  64  feet  was  broken  into  two  lengths  so  that  if 
4 feet  be  taken  from  the  longer  and  added  to  the  shorter,  the  pieces  will  be  equal ; 
what  are  the  lengths  of  the  broken  pieces  ? 

49.  A lady  having  two  watches  bought  a chain  worth  $20.  If  the  chain  be 
put  upon  the  silver  watch,  they  together  will  be  worth  § as  much  as  the  gold 
watch ; if  the  chain  be  placed  upon  the  gold  watch,  the}T  together  will  be  worth 
4 times  as  much  as  the  silver  watch.  What  is  each  watch  worth  ? 

50.  A boat  which  sails  at  the  rate  of  6 miles  an  hour  moves  down  a river 
whose  current  is  3 miles  an  hour;  how  far  can  it  go  and  return  in  8 hours? 

51.  A philanthropist  gave  f of  his  money,  less  $12,  to  a poor  family  and 
then  had  $21  left;  how  much  did  he  give  awa}7? 

52.  A toy  balloon  in  the  air  fell  J of  the  distance  to  the  ground  and  then 
arose  1 the  distance  it  was  from  the  ground ; what  part  of  the  first  distance  is  it 
now  from  the  ground  ? 

53.  If  J of  my  weight  be  added  to  my  weight  and  from  the  sum  55  pounds 
be  taken,  the  result  will  be  200  pounds.;  what  is  my  weight? 

54 ■ If  the  height  of  a monument  be  increased  bv  its  1 and  from  this  sum  be 
taken  the  difference  between  its  J and  ^ , the  result  will  be  40  feet;  what  is  its 
height  ? 

55.  The  sum  of  three  numbers  is  55 ; the  second  is  J greater  than  the  first 
and  the  third  is  twice  the  second  ; find  the  numbers. 

56.  A man  walks  120  miles  in  three  days,  walking  each  day  10  miles  more 
than  the  preceding  day ; how  many  miles  did  he  walk  the  second  day  ? 

57.  Two  pipes  fill  a cistern  of  200  barrels  ; f of  what  flows  through  one  pipe 
being  equal  to  \ of  what  flows  through  the  other  ; how  much  is  carried  in  by 
each  pipe? 


REVIEW  MENTAL  PROBLEMS 


331 


58.  Hunter’s  money  is  to  Fisher’s  money  as  2 to  3;  how  much  money 
has  each,  if  Hunter’s  money  and  J of  Fisher’s  money  is  equal  to  $350? 

59.  If  8 men  can  do  a piece  of  work  in  6 days,  how  many  days  will  be 
needed  to  complete  the  work,  if  4 men  be  added  when  the  work  is  J done? 

60.  Jones  and  Smith  rent  a pasture  for  $40.  Jones  puts  in  12  horses  and 
Smith  8 cows.  How  much  should  each  pay  if  2 cows  eat  as  much  as  3 horses? 

61.  X,  Y and  Z build  a wall,  working  an  equal  number  of  days;  X gets  $3 
a day  ; Y,  $2  a day  and  Z,  $1  a day.  They  receive  $60 ; what  is  each  one’s 
share  ? 

62.  Brown  can  build  a shed  in  10  days  and  Black  in  12  days;  how  many 
days  would  it  take  to  build  it  if  they  work  together? 

63.  White  and  Grey  working  together  can  build  a boat  in  4f  days.  White 
working  alone  will  build  it  in  8 d4ys  ; in  how  many  days  would  Grey  working 
alone  build  it? 

dj.  A piece  of  work  can  be  done  by  4 men  or  6 boys  in  10  days;  how  long 
would  it  require  2 men  and  10  boys  to  do  it  ? 

65.  Wood  can  cut  a cord  of  wood  in  J of  a day  and  Black  can  cut  it  in  f of 
a day  ; working  together,  how  many  cords  can  they  cut  in  a week  of  6 days? 

66.  William  receives  $2  a day  and  pays  50  cents  a day  board  ; at  the  end  of 
40  days  he  has  saved  $40  ; how  many  days  was  he  idle? 

67.  James  had  3 loaves  of  bread  and  David  had  5 loaves,  which  they  shared 
with  Benjamin,  who  gave  them  16  cents;  how  shall  they  share  the  money? 

68.  Two  men  mow  a field  of  grass  for  $40 ; the  first  mows  twice  as  much  as 
the  second,  less  4 acres,  and  is  paid  $24  ; how  many  acres  does  each  mow? 

69.  What  is  the  asking  price  of  cloth,  if  by  dropping  25%  and  selling  it  at 
$2.40  a yard,  a gain  of  20%  is  made  ? 

70.  A sum  of  money  put  on  interest  at  a certain  rate  for  3 years  amounts  to 
$236,  and  the  same  sum  at  the  s ime  rate  for  10  years  amounts  to  $320  ; find  the 
sum  and  rate 

71.  If  5 dollars  be  taken  from  f of  a sum  of  money  there  will  remain  the 
original  sum  ; what  was  the  sum  ? 

72.  A tank  can  be  filled  by  an  inlet  pipe  in  4 hours  and  emptied  when  full 
by  an  outlet  pipe  in  1J  hours ; if  the  tank  be  full  and  both  pipes  opened,  in  what 
time  will  the  tank  be  empty  ? 

73.  A working  alone  can  build  a boat  in  7 days,  B in  8 days,  and  C in  6 
days  ; in  what  time  can  they,  working  together,  build  it? 

7J.  A cyclist  makes  a journey  of  300  miles  in  4 days,  riding  each  day  10 
miles  farther  than  on  the  preceding  day ; how  many  miles  did  he  ride  each  day  ? 

75.  If  5 men  can  do  a piece  of  work  in  a certain  time,  how  many  men  would 
be  required  to  do  double  the  amount  of  work  in  half  the  time  ? 


GENERAL  REVIEW  PROBLEMS 

1087.  1.  The  net  proceeds  of  a sale  are  $940.10.  The  total  expenses  are,  3% 
commission,  $14.10  storage,  $16.50  drayage.  Find  the  amount  of  the  sale. 

2.  I sold  a house  for  $2450  and  gained  14%.  I sold  another  for  the  same 
price  and  lost  14%.  Find  my  net  loss  or  gain. 

3.  I buy  a piano  for  $250.  I want  to  sell  it  so  as  to  gain  15%  after  throw- 
ing off  10%.  What  price  shall  I ask  for  it? 

If..  Change  £142  19s.  Id.  to  French  money. 

5.  What  is  the  cost  in  U.  S.  money  of  900  tons  of  steel,  invoiced  at  £3  8s. 
1 0 d.  a ton  ? 

6.  I sold  a lot  of  goods  for  $760.25,  and  thereby  lost  12%.  At  what  price 
should  the  goods  have  been  marked  to  gain  10%  and  allow  a trade  discount 
of  3%? 

7.  The  net  proceeds  of  a sale  of  hardware  are  $210.50,  the  commission  5%, 
the  other  expenses  $7.42.  Find  the  sales  and  the  amount  of  the  commission. 

8.  An  insurance  company  insures  a house  valued  at  $42000  for  f of  its 
value  at  If  % premium.  This  company  reinsures  f of  its  risk  at  f%.  What 
rate  per  cent,  is  the  first  company  making  on  its  risk? 

9.  A pipe  can  fill  a cistern  in  24  hours  ; another  pipe  in  30  hours.  There 
is  a delivery  pipe  running  from  the  cistern  that  will  empty  it  in  20  hours.  Sup- 
pose the  cistern  to  be  empty,  and  the  three  pipes  opened  at  once;  in  what  time 
will  it  be  filled  ? 

10.  A can  carry  a ton  of  coal  to  the  third  story  of  a building  in  5^  hours,  B 
can  do  the  same  in  6|  hours,  C in  7f  hours.  In  what  time  could  the  three 
together  carry  2f  tons  ? 

11.  An  insurance  policy  is  written  for  an  amount  sufficient  to  cover  an 
insurance  of  $5000  on  a house,  and  24%  premium.  Find  the  face  of  the  policy. 

12.  A can  do  a piece  of  work  in  12  days,  B in  15  days ; if  C "works  with 
them,  they  can  do  the  work  in  4 days.  In  what  time  could  C do  the  work  alone  ? 

13.  If  it  requires  3f  bushels  of  seed  to  sow  an  acre,  how  many  quarts  will 
be  required  to  sow  a lot  110  feet  square? 

H.  A cistern  8 feet  in  diameter,  7 feet  deep,  has  how  many  gallons  of  water 
in  it  when  the  water  is  within  3 inches  of  the  top? 

15.  The  hypotenuse  of  a right-angled  triangle  is  82  feet,  the  base  62  feet. 
Find  the  altitude. 

16.  What  will  it  cost,  at  48  cents  a square  yard,  to  paint  the  dome  of  a hall, 
the  dome  being  in  the  form  of  a hemisphere  40  feet  in  diameter? 

17.  A has  a circular  garden,  and  B a square  one;  each  contains  2 acres.  A 
walk  6 feet  wide  is  made  around  each  garden.  Which  walk  contains  the  greater 
area,  and  how  much?  How  much  greater  would  the  area  of  each  walk  be  if 
it  surrounded  the  2 acres? 


332 


GENERAL  REVIEW  PROBLEMS 


333 


18.  A square  field  containing  12  acres  is  surrounded  by  a tight  board  fence 
8 feet  I iigli.  What  did  the  boards  cost  at  $22  per  M ? 

19.  I buy  muslin  at  6f  cents  a yard  and  mark  it  to  gain  20%,  but  on 
account  of  a defect  I conclude  to  sell  it  at  85%  of  the  marked  price.  What  is 
my  loss  or  gain  on  1000  yards  ? 

20.  A miller  bought  a ton  of  wheat  at  68  cents  a bushel,  which  he  manu- 
factured into  flour.  If  each  bushel  yielded  36f  pounds  and  he  received  $4.10  a 
barrel,  what  is  his  per  cent,  of  gain  or  loss? 

21.  A bin  is  9 feet  5 inches  long,  4 feet  3 inches  wide,  5 feet  8 inches  deep, 
and  is  § full  of  wheat.  How  many  bushels  does  it  contain  and  what  is  the  wheat 
worth  at  68  cents  a bushel  ? 

22.  How  deep  must  a circular  cistern  4 feet  in  diameter  be  to  hold  40  barrels 
of  water  ? 

23.  The  fore  wheel  of  a carriage  is  3 feet  6 inches  in  diameter;  the  hind 
wheel  4 feet  8 inches.  How  many  revolutions  will  the  smaller  one  make  while 
the  larger  one  makes  900  ? 

21^.  An  agent  sold  goods  for  $1395  and  lost  f of  the  cost.  For  how  much 
should  he  have  sold  the  goods  to  have  made  20%  gain? 

25.  A man  lost  $920,  which  was  25%  of  what  he  had  remaining.  What  per 
cent,  of  his  money  did  he  lose? 

26.  A man  bought  four  houses  for  $15000  and  sold  them  as  follows  : on  the 
first  he  gained  20%  ; on  the  second  he  gained  15%  ; on  the  third  he  lost  15% 
and  on  the  fourth  he  lost  4%.  If  he  received  the  same  amount  of  money  for 
each  house,  what  was  the  cost  of  each? 

27.  A coal  dealer  buys  a car  load  of  coal,  35000  pounds,  at  $4.15  a long  ton, 
which  he  retails  at  $5.60  a long  ton.  What  per  cent,  does  he  gain  if  there  is  a 
loss  of  3%  in  handling  the  coal? 

28.  The  minute-hand  of  a clock  is  5 inches  long,  the  hour-hand  3f  inches. 
Over  how  much  more  area  does  the  minute-hand  pass  in  5 hours  than  the  hour- 
hand  ? 

29.  At  what  per  cent,  above  the  manufacturer’s  price  must  a wholesale 
merchant  mark  goods  so  that  he  can  allow  a retailer  a discount  of  30%  and  5% 
and  still  make  a profit  of  25%  ? 

30.  An  article  is  marked  to  gain  33-J%,  but  the  seller  throws  off  ^ and  the 
collector  is  afterwards  paid  15%  for  collecting  the  debt.  What  is  the  percent, 
of  loss  ? 

31.  The  owner  of  a mill  had  it  insured  at  a rate  of  If  %.  He  afterwards 
introduced  steam  power,  and  the  company  took  an  additional  risk  of  $1500. 
They,  however,  raised  the  rate  f%.  The  extra  premium  amounted  to  $65. 
For  what  amount  was  the  mill  first  insured  ? 

32.  A book  jobber  buys  books  at  40%  and  10%  off  list  and  marks  his  stock 
at  an  advance  of  15%  on  list  prices.  He  makes  a sale  to  a private  library  at  a 
discount  of  15%  from  his  marked  price  and  makes  a profit  of  $2500.  How 
much  did  the  books  cost  the  jobber? 


334 


GENERAL  REVIEW  PROBLEMS 


33.  I ordered  an  agent  to  buy  flour  for  me  which  I afterwards  sold  at  20% 
profit  and  gained  $1.50  on  a barrel.  If  my  agent’s  commission  was  3|%,  and 
his  total  commission  $25,  how  many  barrels  did  he  buy? 

34-  A dairyman  sold  a quantity  of  milk,  butter  and  cheese,  receiving  for  all 
$115.25.  He  gained  15%  on  the  milk,  18%  on  the  cheese,  22%  on  the  butter. 
If  the  amount  received  for  each  was  equal,  find  the  cost  of  each. 

35.  A grocer  wishes  to  mix  coffees  worth  38,  42,  45,  50  and  55  cents  a pound 
so  as  to  produce  a mixture  worth  45  cents  a pound.  Find  the  number  of  pounds 
of  each  kind  he  must  take. 

36.  A clerk  is  required  to  mark  goods  so  as  to  make  25%  after  throwing  off 
10%.  By  selling  a lot  of  these  goods  for  $78  net,  there  was  a gain  of  but  3%. 
How  should  he  have  marked  the  goods? 

37.  I go  to  bank  to-day  with  four  notes  : one  dated  to-day,  at  90  days,  for 
$1500  ; the  second  dated  7 days  ago,  at  three  months,  for  $920;  the  third  dated 
to-day,  at  60  days,  for  $1720 ; the  fourth  is  my  own  note  dated  to-day,  at  90  days, 
for  enough  to  raise  my  bank  balance  to  $5000.  What  was  the  face  of  my  own 
note,  my  balance  previous  to  having  these  notes  discounted  being  $290.62  ? 

38.  I buy  in  Paris  328  meters  of  silk  at  27.42  francs  per  meter.  If  exchange 
is  at  5.19,  what  is  the  cost  of  a draft  that  will  pay  for  the  invoice? 

39.  A river  is  moving  at  the  rate  of  4 miles  an  hour.  It  is  119  feet  wide 
and  has  an  average  depth  of  104  feet.  How  many  tons  of  water  will  pass  a 
given  point  on  the  river’s  bank  in  12  minutes,  water  weighing  62J  pounds  to 
the  cubic  foot  ? 

Ifi.  On  May  12,  1908,  I asked  my  broker  to  purchase  for  me  320  shares  Cen- 
tral Railroad  stock  and  left  with  him  as  margin  $3000.  He  bought  on  May  27 
at  1064 ; on  June  2 he  sold  150  shares  at  107f,  and  on  June  18  he  sold  the 
remainder  at  107J.  Our  account  was  settled  on  June  30.  How  much  was  due 
me  on  that  day,  and  what  was  my  gain? 

41.  Two  concentric  circles  have  diameters  of  350  feet  and  390  feet,  respect- 
ively. What  is  the  area  of  the  space  enclosed  between  the  circumferences? 

4%.  A wall  54  feet  high,  2J  feet  thick,  cost  for  building  $2006.90,  at  the  rate 
of  $1.18  a perch.  Find  the  length  of  the  wall. 

43.  A can  do  a piece  of  work  in  27  days ; B in  36  days;  C in  42  days.  They 
all  work  together  until  | of  the  work  is  done,  when  A retires,  leaving  B and  C to 
finish  it.  In  what  time  is  the  whole  work  done?  How  long  do  B and  C work  ? 

44 ■ A ditch  426  feet  long,  24  feet  wide,  54  feet  deep,  cost  for  digging  $1522.65. 
How  wide  should  a ditch  be  that  is  294  feet  long,  4f  feet  deep,  to  cost  $2420  ? 

45.  A cylinder  is  7 feet  in  diameter,  38  feet  long.  Find  its  entire  surface. 

46.  A C3rlindrical  tank  is  7 feet  in  diameter  and  15  feet  deep.  It  has  in  it 
560  gallons  of  water.  What  per  cent,  is  this  of  its  capacity? 

47.  A man  bought  a house  and  lot,  paying  five  times  as  much  for  the  house 
as  for  the  lot.  If  the  house  had  cost  25%  less  than  it  did,  it  would  have  cost 
$3800.  Find  the  cost  of  each. 


GENERAL  REVIEW  PROBLEMS 


335 


4-8.  A dealer  spent  equal  sums  in  hats,  coats  and  shoes,  and  sold  at  a profit 
of  20%  on  the  hats,  22%  on  the  coats,  and  15%  on  the  shoes.  The  entire  sales 
amounted  to  $22500.  Find  the  cost  of  each. 

49.  A invested  a certain  amount  in  real  estate,  and  B invested  4 times  as 
much.  A lost  15%  and  B gained  20%.  The  difference  between  the  amounts 
received  was  $1250.  How  much  did  each  invest? 

50.  A company  engaged  an  agent  to  sell  for  them  on  a commission  of  5% 
and  shipped  him  merchandise  amounting  to  $5500.  The  agent  purchased 
additional  merchandise  to  the  amount  of  $1200,  and  the  firm  sent  him  $500 
cash.  At  the  end  of  a certain  period  the  agent  had  sold  goods  to  the  amount  of 
$4920,  and  the  goods  on  hand  amounted  to  $3006  78.  Find  the  loss  or  gain. 
Find  also  the  amount  due  the  firm  by  the  agent  at  this  time. 

51.  A circular  piece  of  ground,  containing  5 acres,  is  enclosed  by  a tight 
board  fence  7 feet  high.  What  was  the  cost,  at  $13  per  M,  of  1-inch  lumber 
required  to  make  it  ? 

52.  A and  B form  a partnership  on  January  1, 1908,  with  the  understanding 
that  the  gains  and  losses  are  to  be  divided  in  proportion  to  their  average  net 
investments.  On  January  1 A invests  $3000  and  B invests  $2500.  A on 
March  1 invests  $1500,  on  July  1 $2200,  and  withdraws  on  September  1 $2000. 
B on  April  1 invests  $1800  and  on  September  1 $2000,  and  withdraws  on 
November  1 $3300.  A statement  of  their  resources  and  liabilities  at  time  of 
settlement  is  as  follows  : 


Resources : Cash, 

$2200 

Liabilities:  Bills, 

$1920 

Mdse., 

6850 

Sundry  Accounls, 

2600 

Bills, 

3290 

Book  Accounts, 

2600 

Close  each  partner’s  account. 

53.  If  a wall  520  feet  long,  4J  feet  high,  2J  feet  thick,  requires  the  work  of 
175  men,  212  days  of  10  hours  each  in  building,  how  high  should  a wall  be  that 
is  248  feet  long,  2£  feet  thick,  to  require  the  work  of  98  men,  217  days  of  9 hours 
each  ? 

54-  Sold  for  account  of  consignor  on  September  8,  1899,  $5240.26  on  4 
months’  credit;  September  29,  $538  cash;  October  16,  $2750.85,  60  days;  on 
September  15,  paid  freight  amounting  to  $95.23;  my  commission  for  selling 
is  5%.  What  is  due  to  consignor,  and  when  are  the  proceeds  due  by  equation  ? 

55.  A merchant  bought  22  pieces  of  cloth,  each  containing  25  yards,  at 
$4.50,  on  a credit  of  6 months,  and  sold  them  at  $6.25  on  a credit  of  3 months. 
What  was  his  cash  gain,  money  being  worth  6%  ? 

56.  Bought  a check  on  a bank  which  had  suspended,  at  65%  of  its  face,  and 
exchanged  it  for  5%  railroa  l bonds  at  82.  What  rate  of  interest  do  I receive 
on  my  investment  ? 

57.  Bought,  at  8%  discount,  a 6%  mortgage  for  $3800,  with  two  years  to 
run.  What  interest  do  I get  on  the  money  invested,  if  the  mortgage  is  paid  at 
maturity  ? 


336 


GENERAL  REVIEW  PROBLEMS 


58.  A bath  tub  will  hold  200  gallons  of  water.  It  is  filled  by  a faucet 
discharging  50  gallons  in  4 minutes,  and  emptied  by  a waste  pipe  discharging 
48  gallons  in  4 minutes.  If  both  the  pipes  are  opened  at  once  and  the  waste 
pipe  closed  one  hour  afterwards,  in  what  time  will  the  tub  be  full? 

59.  A retail  merchant  sold  a quantity  of  silks  for  $1250,  thereby  gaining 
20%.  The  wholesale  merchant  from  whom  he  bought  them  made  a profit 
of  15%,  and  the  importer  who  sold  them  gained  20%.  What  did  they  cost  the 
importer? 

60.  A merchant  in  Philadelphia  made  the  following  importation  from 
Belfast,  Ireland: 

1720  yds.  linen  at  2s.  6d. 

1850  “ “ “ 3s.  3d. 

512  “ “ “ 5s.  2d. 

1680  “ “ “ 3s.  id. 

Duty  22%  ad  valorem  and  8 cents  a yard  specific.  Find  the  duty.  Find  also 
the  cost  of  a draft  in  payment  of  the  invoice,  exchange  at  4.87. 

61.  A manufacturer  carried  on  business  for  four  years.  The  first  year  he 
gained  20%,  which  at  the  close  of  that  year  he  put  into  the  business.  The  sec- 
ond year  he  lost  15%.  The  third  year  he  gained  25%  of  what  he  had  at  the 
beginning  of  that  year,  and  put  it  into  the  business.  The  fourth  year  he  gained 
16§ %,  and  then  his  total  capital  was  $40250.85.  How  much  had  he  gained  in 
the  four  years? 

62.  If  the  sales  in  a certain  merchandise  account  are  $256742.90,  the  cost  of 
the  merchandise  $275420.82,  and  the  inventory  $45923.75,  what  per  cent,  is  the 
profit? 

63.  The  list  price  of  a sewing  machine  is  $60;  an  agent  buys  at  25%  off 
and  sells  for  $55.  What  is  his  per  cent,  of  profit  ? 

6 If..  A merchant  sells  goods  at  retail  35%  above  cost,  and  at  wholesale  15% 
less  than  the  retail  price.  What  is  his  gain  per  cent,  at  wholesale? 

65.  If  I buy  at  list  less  40%,  will  I gain  or  lose  if  I add  50%  to  the  list  and 
sell  at  | off? 

66.  What  is  the  balance  due  June  15, 1909,  on  a note  of  $1350,  dated  June 
15,  1908,  and  showing  the  following  indorsements : August  18,  1908,  $50  ; Sep- 
tember 23,  1908,  $75;  October  20,  1908,  $90;  November  24,  1908,  $75  ; January 
14,  1909,  $5:  March  25,  1909,  $60;  April  21,  1909,  $350?  Work  by  the  United 
States  rule. 

67.  Find  the  cost  of  300  shares  Reading  Railroad  stock  (par  $50)  at  Ilf, 
brokerage  f %. 

68.  I bought  290  shares  Philadelphia  Traction  at  72f,  brokerage  \%. 
What  did  they  cost  me? 

69.  I bought  a bill  of  exchange  on  London,  and  paid  $2892.50  for  it.  What 
was  the  face  of  the  bill,  exchange  at  4.87f  ? 

70.  What  is  the  face  of  a bill  of  exchange  on  Paris  which  can  be  bought 
for  $1590.62  at  5.15? 


GENERAL  REVIEW  PROBLEMS 


387 


71.  What  is  the  duty  on  9000  yards  of  28-inch  dress  goods  at  42  cents  per 
square  yard  ? 

79.  What  is  the  dut}r  on  720  gallons  of  brandy  at  $1.25  per  gallon,  the 
allowance  for  leakage  being  2%  ? 

73.  A firm  in  Philadelphia  imports  from  Paris  9200  yards  of  22-inch  silk, 
the  marked  price  of  which  at  the  time  of  purchase  was  3.25  francs  per  meter. 
What  was  the  duty  at  10  cents  per  square  yard  and  35%  ad  valorem  ? 

7L.  What  is  the  duty  on  a consignment  of  iron  invoiced  at  £2265  18s.  6d. 
at  42  % ? 

75.  A man’s  personal  property  is  assessed  at  $5000,  his  real  estate  at  $13000. 
Find  his  total  tax  at  the  rate  of  $1.32  per  $100. 

76.  The  expenses  of  a town  for  a year,  not  including  the  tax  collector’s 
commission  of  3%,  are  as  follows:  for  schools,  $3500  ; for  roads,  bridges,  etc., 
$1700;  for  incidental  expenses,  $800.  If  the  total  valuation  of  the  town  is 
$2450000,  what  should  be  A’s  tax  whose  property  is  assessed  at  $3500  ? 

77.  A lumber  dealer  purchased  two  piles  of  wood  ; one  pile  was  66  feet 
long,  7 feet  high,  12  feet  wide,  and  contained  43^-  cords.  The  other  pile  was  29 
feet  long,  16  feet  wide  and  of  sufficient  height  to  contain  54f  cords.  Find  the 
height  of  the  latter  pile. 

78.  A contractor  engaged  to  dig  two  cellars  at  the  same  price  per  cubic 
yard.  He  received  $450  for  one  75  feet  long,  28  feet  wide,  9 feet  deep.  What 
should  he  have  received  for  the  other,  if  its  length  was  88  feet,  width  33  feet, 
depth  8 feet? 

79.  If  52  men  can  earn  $1850  in  19  days  by  working  10  hours  a day,  how 
much  should  29  men  earn  in  23  days  of  8 hours  each? 

80.  If  the  cost  of  pasturing  85  head  of  cattle  for  13  weeks  is  $75,  what 
should  be  the  cost  of  pasturing  99  head  of  cattle  for  4 weeks? 

81.  A note  of  $5000,  dated  July  7,  1908,  with  interest  at  5J%,  has  the  fol- 
lowing indorsements:  September  15,  1908,  $10;  October  28,  1908,  $25;  Decem- 
ber 23,  1908,  $1000!;  February  25,  1909,  $1500;  April  28,  1909,  $90;  May  26, 
1909,  $1300.  What  is  due  July  7, 1909  ? Use  United  States  and  Mercantile  rules. 

82.  I bought  357  shares  Lehigh  Navigation  (par  $50)  at  41|.  brokerage 
Find  the  cost. 

83.  A legacy  of  $15000  is  invested  as  follows : 50  shares  P.  R.  R.  (par  $50) 
at  53f ; 92  shares  Central  at  82J ; 36  Lehigh  Valley  (par  $50)  at  27f ; and  the 
balance  in  Union  Pacific  at  41f.  Brokerage  at  Philadelphia  Stock  Exchange 
rates.  Find  the  number  of  shares  of  Union  Pacific  purchased  and  the  balance 
unspent  (no  fraction  of  a share  being  purchased). 

84-  A fire  insurance  company  had  a risk  of  $58000  at  § % premium,  and 
reinsured -§- of  the  risk  in  another  company  at  f%  premium  and  £ of  it  in 
another  at  f%  premium.  What  rate  per  cent,  is  the  first  company  making  on 
its  risk? 

85.  If  the  premium  paid  for  insuring  a property  was  $45.20,  and  the  rate 
of  insurance  was  sixty  cents  on  $100,  for  what  sum  was  it  insured  ? 


338 


GENERAL  REVIEW  PROBLEMS 


86.  If  from  220  reams  of  paper  3000  copies  of  a book  containing  380  pages 
can  be  printed,  bow  many  reams  will  be  required  to  print  8000  copies  of  a book 
containing  480  pages? 

87.  A grocer  wishes  to  mix  500  pounds  of  coffee  which  be  can  sell  at  75 
cents  a pound  and  gain  25%,  by  using  coffees  worth  38,  40,  45,  65  and  70  cents 
a pound.  How  many  pounds  of  each  should  he  take? 

88.  What  is  the  amount  of  $998.65  from  January  16,  1908,  to  September 
14,  1908?  (Time  by  compound  subtraction  ) 

89.  What  is  the  cost  of  a circular  piece  of  ground  228  feet  in  diameter,  at 
$300  an  acre  ? 

90.  A legacy  of  $22000  is  invested  as  follows:  $2500  of  4%  Government 
Bonds  at  lllf;  $3200  of  6%  railroad  bonds  at  104J ; $1600  of  5%  railroad 
bonds  at  98f ; and  the  balance  in  U.  S.  5s  at  128f.  Find  the  total  income  from 
the  investment.  Find  also  the  balance  unspent  (no  bond  being  of  smaller 
denomination  than  $100),  brokerage  at  Philadelphia  Stock  Exchange  rates. 

91.  If  I buy  4%  stock  at  97f,  brokerage  what  per  cent,  income  do  I 
receive  on  my  investment? 

92.  If  I buy  6%  bonds  at  97f,  brokerage  \% , what  per  cent,  income  do  I 
receive  on  my  investment? 

93.  I bought  a 5%  bond  (par  $1000)  at  such  a price  as  to  yield  me  4J%  on 
my  investment.  Find  the  cost  of  the  bond. 

94-  Bank  stock  paying  5 % is  selling  at  1124.  Find  the  per  cent,  income  on 
investment. 

95.  A stone  wall  3 feet  thick,  44  feet  high,  cost  for  its  construction  $1590, 
at  96  cents  a perch.  Find  the  length  of  the  wall. 

96.  A contractor  agrees  to  build  a cellar  which  shall  be  18  feet  long,  16 
feet  wide  and  9 feet  deep,  inside  measure,  when  finished.  His  contract  price  is 
62  cents  per  cubic  yard  for  making  the  excavation,  and  $1.25  a perch  for  the 
wall.  If  the  wall  is  2 feet  thick,  find  his  total  contract  price. 

97.  At  $18.50  per  M,  what  is  the  cost  of  sufficient  lumber  2 inches  thick  to 
make  50  boxes,  each  4X4  feet,  3 feet  deep,  inside  measure? 

98.  The  expenses  of  a town  for  a year,  not  including  the  tax  collector's  com- 
mission of  3%,  are  as  follows:  for  schools,  $2500  ; for  roads,  bridges,  etc.,  $1200  : 
for  incidental  expenses,  $900.  If  the  total  valuation  of  the  town  is  $1420000, 
what  should  be  A’s  tax  whose  property  is  assessed  at  $5500? 

99.  A fire  insurance  company  had  a risk  of  $85000  at  f % premium,  and 
reinsured  j of  the  risk  in  another  company  at  f-%  premium  and  4 °f  it  in 
another  at  f % premium.  What  rate  per  cent,  is  the  first  company  making  on 
its  risk  ? 

100.  The  stock  of  a wholesale  house  is  insured  in  a number  of  companies 
for  $175000,  and  is  damaged  bv  water  to  the  amount  of  $57641.73.  What  per 
cent,  of  its  risk  should  be  paid  by  each  company,  and  what  amount  should  a 
company  pay  that  had  a risk  of  $2500  upon  it? 


GENERAL  REVIEW  PROBLEMS 


339 


101.  The  value  of  a ship’s  cargo  is  $15500,  and  the  owner  desires  to  insure 
it  for  a sum  which  will  cover  the  value  of  the  cargo  and  the  cost  of  insurance. 
If  the  rate  of  insurance  is  lg-%,  what  should  be  the  amount  of  the  policy? 

102.  A roof  is  42  feet  long,  28  feet  from  eaves  to  ridge.  How  many  slates 
are  required  to  cover  it,  if  the  slates  used  are  10  inches  wide,  showing  12  inches 
to  the  weather,  and  what  will  they  cost  at  $10.90  a square? 

Note.— A square  contains  100  square  feet. 

103.  Find  the  entire  surface  of  a cylinder  9 feet  in  diameter,  15  feet  long. 

101/..  A garden  125  feet  long,  86  feet  wide,  has  on  its  border  a stone  wall 

3 feet  high,  18  inches  thick.  What  is  the  cost  of  the  wall  at  92  cents  a perch? 

105.  If  the  premium  paid  for  insuring  a property  was  $45.20,  and  the  rate 
of  insurance  was  sixty  cents  on  $100,  for  what  sum  was  it  insured  ? 

106.  The  stock  of  a wholesale  house  is  insured  in  a number  of  companies 
for  $15000,  and  is  damaged  bv  water  to  the  amount  of  $900.  What  per  cent,  of 
its  risk  should  be  paid  by  each  company,  and  what  amount  should  a company 
pay  that  had  a risk  of  $6500  upon  it? 

107.  The  value  of  a ship’s  cargo  is  $20750,  and  the  owner  desires  to  insure 
it  for  a sum  which  will  cover  the  value  of  the  cargo  and  the  cost  of  insurance. 
If  the  rate  of  insurance  is  1|%,  what  should  be  the  amount  of  the  policy? 

108.  A lumber  dealer  purchased  two  piles  of  wood ; one  pile  was  72  feet 
long,  10  feet  wide,  and  high  enough  to  contain  45  cords.  The  other  pile  was  18 
feet  wide,  16  feet  high,  and  long  enough  to  contain  87f  cords.  Find  the  height 
of  the  first  pile  and  the  length  of  the  second  pile. 

109.  A contractor  engaged  to  dig  two  cellars  at  the  same  price  per  cubic 
yard.  He  received  $350  for  one  70  feet  long,  25  feet  wide,  8 feet  deep.  What 
should  he  have  received  for  the  other,  if  its  length  was  96  feet,  width  32  feet, 
depth  9 feet? 

110.  If  42  men  can  earn  $1580  in  18  days  by  working  11  hours  a day,  how 
much  should  19  men  earn  in  29  days  of  9 hours  each? 

111.  If  the  cost  of  pasturing  75  head  of  cattle  for  11  weeks  is  $72,  what 
should  be  the  cost  of  pasturing  92  head  of  cattle  for  3 weeks? 

112.  I go  to  bank  to-day  with  two  notes,  one  dated  seven  days  ago  for  $2200, 
at  3 months;  the  second  is  my  own  note  dated  to-day,  at  90  days,  for  a sum 
sufficient  to  make  my  balance  large  enough  to  enable  me  to  give  a check  for 
$4500  and  have  a balance  of  $300  in  bank.  If  my  present  balance  is  $290,  what 
is  the  face  of  the  note? 

113.  If  it  costs  $19.20  to  carry  7642  pounds  129  miles,  what  should  be  the 
cost  of  carrying  12693  pounds  68  miles  at  the  same  rate? 

114..  85%  of  a meter  is  what  per  cent,  of  a yard  ? 

115.  I have  in  bank  to-day  $311.72.  I send  to  bank  H.  R.  Reed’s  note  dated 
to-day  for  $726  at  3 months,  and  my  own  note  dated  to-day  at  90  days  for 
a sum  sufficient  to  enable  me  to  give  a check  for  $1150  and  have  left  in  bank 
$125.10.  Find  the  face  of  my  note. 


340 


GENERAL  REVIEW  PROBLEMS 


116.  In  the  following  account  state  when  the  balance  is  due  by  equation, 
and  what  amount  of  cash  will  settle  it  June  1,  1909. 

Dr.  Frank  B.  Hughes  Cr. 


1908 

Mar. 

2 

Balance 

911 

72 

1908 

Mar. 

27 

Cash 

900 

00 

U 

28 

Mdse.,  60  days 

375 

93 

Apr. 

22 

Note,  10  days 

370 

00 

Apr. 

9 

“ 2 mos. 

702 

10 

May 

9 

Draft,  30  days 

600 

00 

June 

16 

“ 1 mo. 

397 

98 

117 . If  19  men  can  do  a piece  of  work  in  12  days,  how  many  days  should 
26  men  require  to  do  the  same  work? 

118.  If  14  men  or  19  boys  can  do  a piece  of  work  in  42  days,  in  what  time 
can  12  men  and  22  boys  do  the  same  work? 

119.  A can  do  a piece  of  work  in  9 days,  B in  9J  days,  and  C in  10J  days. 
In  what  time  could  they  do  it,  working  together? 

120.  I send  to  bank  to-day  William  Smith’s  note  dated  to-day,  in  my  favor, 
for  $950  at  three  months,  and  my  own  note  at  60  days,  dated  to-day,  for  a sum 
sufficient  with  the  proceeds  of  Smith’s  note  to  allow  me  to  draw  a check  for 
$1450  and  leave  in  bank  a balance  of  $375.50.  If  my  balance  before  sending 
these  notes  to  bank  for  discount  was  $250.10,  what  is  the  face  of  my  own  note  ? 

121.  A’s  money  is  5%  less  than  B’s,  and  B’s  is  15%  greater  than  C’s.  If 
they  together  have  $19360.72,  how  much  does  each  have? 

122.  I send  an  agent  $1190  to  invest  in  apples  at  $1.10  a barrel,  allowing 
him  a commission  of  4%.  He  pays  17  cents  apiece  for  barrels  and  22  cents  a 
barrel  transportation.  How  many  barrels  of  apples  does  he  buy  ? What  is  the 
surplus  ? 

123.  On  a lot  of  merchandise  which  I sold  for  $962.50,  I lost  18%.  The 
cost  price  of  the  remaining  merchandise  is  $2642.50.  At  what  per  cent,  advance 
must  I sell  this  to  recover  my  loss? 

124-  In  what  time,  at  simple  interest,  will  $1796.14  amount  to  $2496.12 
at  4i%  ? 

125.  What  amount  will  settle  the  following  account  November  11,  1908? 

Dr.  H.  M.  Kennedy  Cr. 


1908 

June 

6 

Balance 

403 

19 

1908 

July 

13 

Cash 

475 

U 

22 

Mdse.,  60  days 

1126 

92 

Sept. 

18 

Note,  10  days 

500 

July 

26 

“ 2 mos. 

472 

85 

ov. 

4 

Draft,  30  “ 

375 

Aug. 

19 

1 mo. 

396 

12 

126.  A rectangular  field  produces  1726  bushels  of  wheat,  at  the  rate  of  82 
bushels  to  the  acre.  If  one  side  of  the  field  is  726  feet,  what  is  the  distance 
around  the  field  ? 

127.  How  many  bricks  8 inches  long  and  4 inches  wide,  are  required  to  lay 
a pavement  176  feet  long,  9 feet  wide?  What  will  they  cost  at  $13.50  a 
thousand. 


GENERAL  REVIEW  PROBLEMS 


341 


128.  A can  do  a piece  of  work  in  10J  days,  B in  11 J days,  C in  12 \ days. 
In  what  time  could  they  do  the  work  together? 

129.  An  importer  buys  in  France  426  meters  of  silk  at  26.14  francs  per 
meter.  He  pays  40%  duty.  What  should  the  entire  quantity  be  marked  in 
order  to  gain  15%  and  throw  off  5%  ? 

130.  I buy  two  houses  for  $9670,  giving  14%  more  for  one  than  for  the  other. 
How  should  I sell  the  cheaper  house  to  gain  20%  ? 

131.  What  is  the  duty  at  42%  on  456  dozen  pocket  knives  invoiced  at  17s. 
4d.  a dozen? 

132.  The  net  proceeds  of  a consignment  are  $1496.72.  The  rate  of  commis- 
sion is4%.  The  other  expenses  are : $14.62,  drayage ; $26.75,  storage;  $32.96, 
sundry  expenses  ; what  were  the  total  sales  ? 

133.  At  what  price  per  yard  must  I sell  silk  that  cost  me  5.42  francs  a meter, 
in  order  to  gain  30%  ? 

131/..  Certain  cloth  shrinks  5%  of  its  length  in  sponging.  How  many  yards 
of  the  cloth  did  I have  before  sponging,  if  after  sponging  there  are  672J  yards  ? 

135.  I purchased  a quantity  of  green  coffee.  In  roasting  it  there  is  a loss 
of  10%  of  its  weight.  What  was  the  gross  cost  of  the  coffee,  purchased  at  18 
cents  a pound  green,  if  after  roasting  I sell  the  entire  quantity  for  $288  at  24 
cents  a pound  ? 

136.  I import  from  Sheffield,  Eng.,  the  following : 120  dozen  knives  at  18s. 
3d.;  215  dozen  knives  at  £l  Is.  Id. ; 295  dozen  knives  at  £1  3s.  2d. ; 82  dozen 
knives  at  £1  7s.  8 d.  I pay  40%  duty,  £1  19s.  2d.  consul  fees,  15s.  3d.  drayage, 
3%  commission  for  buying,  and  $7.90  drayage  in  Philadelphia.  Find  the  total 
cost,  exchange  at  4.86J. 

137.  If  220  men  working  9 hours  a day  can  build  a wall  550  feet  long,  31- 
feet  high,  2^  feet  thick  in  268  days,  how  long  a wall  should  185  men,  working 
10  hours  a day,  build,  if  the  wall  is  7J  feet  high,  2J  feet  thick,  and  the  men  work 
275  days  ? 

138.  On  June  19,  1908,  I give  a note  payable  on  demand  for  $2800  with 
interest  at  4%.  I pay  $500  each  quarter.  What  amount  will  take  up  the  note 
one  year  from  date,  there  having  been  three  payments  made?  Use  both  United 
States  and  Mercantile  rules. 

139.  A can  do  a piece  of  work  in  15  days,  B in  18  days,  C in  20  days.  The 
three  work  together  until  -fa  of  the  work  is  done,  when  A retires,  leaving  B and 
C to  finish  it.  In  what  time  will  the  work  be  done? 

11/0.  A man  invests  $25000  as  follows:  72  shares  of  5%  stock  at  143| ; 27 

shares  of  4%  stock  at  105| ; 22  shares  of  5%  stock  at  109f  ; and  the  balance  in 
stock  paying  6%  at  115J.  What  is  his  total  income,  brokerage  \ % ? 

11/1.  What  is  the  weight  of  an  iron  pipe  18  feet  long,  the  inside  diameter 
being  10  inches  and  the  outside  diameter  12  inches,  if  a cubic  foot  of  iron  weighs 
450  pounds  ? 

11/2.  A field  contains  10  acres;  it  is  in  the  form  of  a square.  Find  the 
length  of  the  diagonal  of  the  field. 


342 


GENERAL  REVIEW  PROBLEMS 


llf.3.  A ball  9 inches  in  diameter  weighs  750  pounds.  How  many  balls  of 
the  same  metal,  three  inches  in  diameter,  will  be  required  to  weigh  as  much  ? 

In  the  following  statement,  when  is  the  balance  due  by  equation? 
What  amount  would  settle  it  September  28,  1908? 


Dr.  John  D.  Bartram  Gr. 


1908 

Mar. 

5 

Balance 

492 

50 

1908 

Mar. 

16 

Cash 

150 

00 

U 

24 

Mdse.,  60  days 

450 

92 

Apr. 

22 

Note,  4 months 

350 

00 

Apr. 

15 

“ 2 mos. 

490 

15 

May 

27 

Draft,  90  days 

380 

00 

U 

22 

“ 10  days 

690 

15 

Sept. 

22 

Cash 

270 

00 

May 

18 

“ 1 mo. 

212 

42 

1 

lJt.5.  On  June  22,  1908,  I gave  William  Hafler  a note  for  $1800  and  agreed 
to  pay  $150  every  Jay  thereafter  until  the  29th  of  June,  inclusive,  when  I would 
give  a new  note  for  the  balance  due  on  that  day.  If  the  interest  is  at  6%,  what 
will  be  the  face  of  the  new  note?  United  States  rule. 

llf.6.  What  is  the  value  of  a shaft  of  iron  18  inches  in  diameter  and  22  feet 
long,  at  $16.50  a ton  (2240  lb.),  a cubic  foot  of  iron  weighing  450  pounds? 

llf.7 . A and  B form  a partnership,  each  putting  in  $3000.  At  the  end  of  3 
months  A draws  out  $1500  and  B $400 ; at  the  end  of  6 months  A draws  out 
$300  and  B $200 ; and  at  the  end  of  9 months  A draws  out  $750  and  B $650.  At 
the  end  of  the  year  they  dissolve  partnership,  having  a capital  of  $3220.  How 
must  they  divide  it? 

llf.8.  Reed  & Co.  fail  in  business  with  liabilities  amounting  to  $65650.  The 
assignees  sold  the  real  estate  for  $19500  and  the  stock  of  goods  for  $2700;  col- 
lected debts  owing  them  amounting  to  $7200,  and  expended  $1250  in  settling  up 
the  business.  What  will  William  Harper  receive  if  his  claim  is  $12720? 

lJf.9.  The  above-mentioned  firm  8 years  afterwards  met  with  sufficient  suc- 
cess to  enable  them  to  pay  off  their  debts  in  full,  with  interest  at  5%.  What 
would  John  Baker  receive  now,  if  his  original  claim  was  $7250? 

150.  A retail  dealer’s  private  mark  is  “ Blacksmith  ”.  How  does  he  mark 
goods  which  he  buys  for  $2.40  and  sells  at  a gain  of  25%  ? 

Note. — The  letters  of  the  private  mark,  in  their  respective  order,  correspond  to  the  digits  1,  2, 
3,  4,  5,  6,  7,  8,  9 and  0. 

151.  A retail  dealer’s  private  mark  is  “ Charleston.”  He  has  marked  a lot 
of  goods  at  $h.lr  per  yard.  Find  the  cost  of  the  goods,  if  his  gain  is  20%. 

152.  On  June  1,  1908,  I purchased  through  a broker  ISO  shares  of  Lake 
Shore  R.  R.  at  84J  and  deposited  $2000  as  margin.  On  June  18,  he  sold  75 
shares  at  87§,  and  on  June  26,  he  sold  the  balance  for  88R  On  June  15.  a 
dividend  of  4J%  was  declared.  What  is  due  me  on  June  30?  What  is  my 
profit? 

153.  A and  B are  equal  partners  in  business,  and  at  the  time  of  settlement 
it  is  found  that  A has  to  his  credit  $4250.75,  while  B has  overdrawn  his  account 
$1920.65.  There  are  no  other  resources  or  liabilities.  How  shall  they  settle? 


GENERAL  REVIEW  PROBLEMS 


343 


15 If..  John  R.  Lamper  and  R.  E.  Butterwick  are  masons  and  form  a partner- 
ship to  do  job-work.  They  agree  to  share  equally.  Each  partner  keeps  an 
account  of  the  labor  he  has  performed,  the  expenses  he  has  paid  and  the  collec- 
tions he  has  made.  Lamper’s  labor  amounted  to  $2850.62.  Butterwick’s  labor 
amounted  to  $3002.65.  Lamper  paid  for  expenses  $196.15,  but  lost  10%  of  his 
labor  accounts.  Butterwick  paid  for  expenses  $212.62,  and  failed  to  collect 
$302.25.  Adjust  each  partner’s  account. 

155.  John  Elarper  begins  a business  January  1,  1908,  which  is  to  consist  of 
preparing  building  lumber.  He  rents  a factory  for  $300  a year.  He  bu}Ts  raw 
lumber  amounting  to  $2550  and  sells  during  the  year  finished  lumber  to  the 
amount  of  $1475.22.  He  has  paid  $217.32  during  the  year  for  necessary 
expenses,  and  for  labor  $725.38.  Find  his  loss  or  gain  for  the  year,  if  the  material 
on  hand  inventories  $2175. 

156.  James  Gorvin,  of  Kansas,  and  Henry  Trexler,  of  Allentown,  Pa., 
engaged  in  buying  and  selling  western  horses.  Trexler  advanced  to  Gorvin 
$5200  during  a certain  period,  and  Gorvin  purchased  horses  to  the  amount  of 
$9250.  At  one  time  Gorvin  shipped  to  Trexler  horses  valued  at  $4250,  and 
at  another  time  horses  valued  at  $8720.  During  the  period  Trexler  made 
sales  amounting  to  $6290,  and  Gorvin  made  sales  amounting  to  $2420.  Trexler 
spent  for  expenses  incident  to  the  business  $290.68,  and  Gorvin  $320.75. 
They  agree  to  discontinue  the  partnership,  and  from  an  inventory  taken  it  is 
found  that  Gorvin  has  horses  on  hand  valued  at  $1720,  and  Trexler  has  horses 
on  hand  valued  at  $2950.  If,  as  a part  of  the  settlement,  each  agrees  to  take  the 
horses  he  has  on  hand,  find  their  loss  or  gain  and  how  much  is  due  one  from  the 
other. 

157 . A and  B united  in  the  purchase  of  two  car  loads  of  coal  to  be  divided 
between  them.  The  invoice  was  as  follows: 


Date 

Car  No. 

Egg 

Stove 

Price 

Total 

Tods 

cwt. 

Tons 

cwt. 

% 

cts. 

% 

cts. 

July  9 

U 

44303 

47363 

20 

00 

20 

00 

2 

2 

65 

75 

53 

55 

10S 

00 

00 

~oo 

The  freight  was  $1.70  a ton  on  40  tons.  The  coal  was  weighed  by  the 
wagon  load  as  delivered.  A received  23625  lbs.  of  the  egg  coal  and  23375  lbs.  of 
the  stove  coal ; B received  22000  lbs.  of  the  egg  coal  and  21900  lbs.  of  the  stove 
coal.  In  settling  for  the  coal  and  freight,  A paid  $91  and  B paid  $85  ; was  this 
settlement  correct  ? 

158.  If  15%  of  a certain  number  is  2232  more  than  3%  of  three  times  the 
number,  what  is  the  number? 

159.  Received  an  invoice  of  goods,  14%  of  which  was  unsalable ; at  what 
per  cent,  above  cost  must  I sell  the  remainder  in  order  to  clear  30%  on  the 
whole  invoice? 


344 


GENERAL  REVIEW  PROBLEMS 


160.  Find  the  net  amount  of  the  following  invoice  of  glassware: 

25  doz.  goblets  @ 60  cents,  40%  and  10%  off; 

40  doz.  goblets  @ 55  cents,  35%  and  5%  off ; 

50  doz.  dishes  @ 85  cents,  50%  and  15%  off; 

60  doz.  dishes  @ $1.25,  30%  and  3%  off. 

161.  A Georgia  planter  shipped  400  bales  of  cotton,  averaging  510  lb.  each, 
to  a Philadelphia  broker  who  sold  it  for  him  at  8J  cents  a pound,  charging  3% 
commission  and  $73.40  for  other  expenses.  The  broker  remitted  the  proceeds 
by  draft,  purchased  at  |%  premium  ; what  was  the  face  of  the  draft? 

162.  The  International  Navigation  Company  insured  a vessel  and  cargo 
for  f of  their  value  with  the  London  Marine  Insurance  Company  at  3J%.  The 
London  Marine  reinsured  | of  the  risk  with  the  Transatlantic  Insurance  Com- 
pany at  2f  %.  The  vessel  was  lost  at  sea,  and  the  loss  of  the  Transatlantic  Com- 
pany was  $1200  more  than  that  of  the  London  Marine.  What  amount  did  the 
International  Navigation  Company  lose? 

163.  A merchant  bought  goods  as  follows:  Jan.  1,  1908,  $2234.76;  Feb.  1, 
1908,  $1362.84;  Mar.  15,  1908,  $6327;  May  15,  1908,  $7415.80;  July  1,  1908, 
$4225.66.  On  each  of  these  bills  he  was  entitled  to  a credit  of  four  months. 
What  does  he  owe  December  1,  1908? 

161,..  William  Hendrickson  bought  of  Levi  Morris  merchandise  amounting 
to  $6324.78,  terms  30  days,  and  gave  in  payment  his  note  at  90  days;  what 
was  the  face  of  the  note? 

165.  A note  for  $5000,  with  interest  at  4%,  dated  Jan.  6,  1908,  has  the 
following  indorsements:  Apr.  1,  1908,  $1000;  June  1,  1908,  $1000;  Aug.  12, 
1908,  $1000  ; Oct.  1,  1908,  $1000.  What  is  due  December  1,  1908,  by  the  Mer- 
cantile rule? 

166.  A can  do  a piece  of  work  in  14  days,  B can  do  it  in  20  days.  If  C 
works  with  them,  all  three  can  do  the  work  in  6T4T  days.  A commences  the 
work  and  continues  for  3 days;  B then  relieves  him,  and  works  5 days;  B 
and  C work  together  for  the  next  two  days.  How  long  does  it  then  take  A and 
B to  finish  the  work  ? 

167.  On  January  8,  1906,  I bought  $12000  worth  of  6%  railroad  bonds  at 
116J,  brokerage  \%.  On  May  1 and  November  1 of  each  year  I cashed  the 
coupons,  and  on  September  8, 1908, 1 sold  the  bonds  at  114|,  brokerage  \ %.  How 
much  less  did  I receive  than  if  I had  loaned  my  money  on  mortgage  at  44%  ? 

168.  A steamboat  that  can  run  18  miles  an  hour  with  the  current,  and  14 
miles  an  hour  against  it,  requires  16  hours  to  go  to  a certain  point  and  return. 
What  is  the  distance? 

169.  A Philadelphia  firm  of  exporters  and  importers,  having  received  an 
account  sales  from  its  agents  in  London  showing  net  proceeds  £784  14s.  lOd. 
to  its  credit,  directs  the  agents  to  remit  a draft  for  10248.75  francs  to  one  of  the 
firm’s  creditors  in  Paris,  and  forward  the  balance  by  draft  on  Philadelphia.  If 
Paris  exchange  is  25.18  at  London,  and  Philadelphia  exchange  4.89,  find  the  face 
of  each  draft. 


GENERAL  REVIEW  PROBLEMS 


345 


170.  If  it  requires  125  reams  of  paper  to  print  5000  copies  of  a book  of  280 
pages,  each  page  being  5J  X 8J  inches,  how  many  reams  will  be  required  to 
print  10000  copies  of  a book  of  176  pages,  each  page  6J  X 9J  inches? 

171.  Henkel  & Co.  made  an  assignment  for  the  benefit  of  their  creditors, 
with  the  following  resources  and  liabilities:  Resources. — Cash,  $3263.17;  Mdse., 
$17324.80  ; Real  Estate,  $12000  ; Bills  Receivable,  $26340  ; Accounts  Receivable, 
$32725.62.  Liabilities. — Bills  Payable,  $22637.50;  Jones  & Hoyt,  $25784.32; 
Clark  & Co.,  $18737.90 ; Hanson  & Adams,  $34859.18 ; William  E.  Davis, 
$6284.38 ; Henry  Thompson,  $19374.62 ; Lavelle  & Wilson  Co.,  $27136.40 ; 
Morton  & Haskell,  $7929.82.  The  expenses  of  the  assignment  were  $3214.85. 
Show  how  much  each  creditor  receives. 

172.  Robert  F.  Allison  and  Peter  J.  Crosby  form  a copartnership,  with  the 
understanding  that  each  is  to  share  in  gains  or  losses  in  proportion  to  his  net 
investment.  Allison  invests  $10000,  January  1,  1908;  $3000,  March  10  ; $4000, 
June  1 ; and  $2000,  September  1.  He  withdraws  $475,  February  1 ; $350,  April  1 ; 
$275,  May  1 ; $500,  July  1 ; and  $675,  October  1.  Crosby  invests  $8000,  January 
1,  1908  ; $2000,  April  1;  $1000,  July  1 ; and  $1500,  August  1.  He  withdraws 
$700,  March  1 ; $300,  June  1 ; $200,  August  15  ; and  $150,  September  1.  They 
gain  during  the  year  $3200;  how  should  this  be  divided  on  January  1,  1909? 

173.  Six  per  cent,  interest  is  charged  and  allowed  on  each  item  in  the  fol- 
lowing account.  What  is  the  balance  due  December  31,  1908? 


Dr.  Arthur  C.  Bennett  Cr. 


1908 

Jan. 

14 

Mdse., 

net 

3418 

20 

1908 

Feb. 

19 

Cash 

200000 

Mar. 

4 

U 

60  days 

1725 

44 

May 

20 

Note,  30  days 

1500  00 

Apr. 

21 

(C 

30  “ 

378 

19 

July 

15 

Cash 

1000:00 

Aug. 

10 

u 

60  “ 

2621 

80 

Sept. 

24 

Cash 

2500,00 

17 If..  What  is  the  value  of  a wedge  of  fine  gold  8 inches  long,  5 inches  wide 
and  4 inches  thick  at  the  butt  (the  other  end  being  a sharp  edge)?  The  specific 
gravity  of  gold  is  19.26;  a cubic  foot  of  water  weighs  1000  ounces  avoirdupois; 
one  pound  avoirdupois  equals  7000  grains  Troy  ; and  23.22  grains  of  fine  gold 
are  worth  $1. 

Note. — The  specific  gravity  of  any  solid  or  liquid  denotes  the  ratio  of  its  weight  to  the  weight 
of  an  equal  volume  of  water. 

175.  The  duty  of  40%  on  a quantity  of  silk  imported  from  France,  invoiced 
at  6 francs  a yard,  was  $520.80 ; how  many  yards  were  there  in  the  importation? 

176.  A farmer  bought  one  hundred  animals  for  $100,  consisting  of  calves  at 
$10,  sheep  at  $3,  and  hens  at  50  cents  each.  How  many  of  each  did  he  buy? 

177.  What  will  it  cost,  at  34  cents  a cubic  yard,  to  make  the  excavation  for 
a circular  pond,  50  feet  in  diameter,  2 feet  deep  at  the  circumference  and  5 feet 
deep  at  the  center,  the  bottom  having  a uniform  slope  from  the  circumference 
to  a point  at  the  center  (in  the  form  of  an  inverted  cone)  ? 


346 


GENERAL  REVIEW  PROBLEMS 


178.  If  the  pond  in  Problem  177  be  filled  within  six  inches  of  the  top 
what  per  cent,  of  its  entire  capacity  will  it  contain? 

179.  An  importer  buys  an  invoice  of  Italian  goods,  amounting  to  23724  lire, 
pays  the  duty  at  30%,  and  marks  the  goods  at  25%  above  cost.  He  sells  one 
half  of  the  invoice  at  this  price,  but  is  obliged  to  sell  the  remainder  at  10% 
below  his  marked  price.  Find  his  gain  on  the  entire  invoice. 

180.  A stock  of  goods  worth  $24000  is  insured  in  four  different  companies 
as  follows:  in  the  first  for  $5000;  in  the  second  for  $5000;  in  the  third  for  $4000, 
and  in  the  fourth  for  $3000.  Each  of  the  policies  contains  the  80%  co-insurance 
clause.  If  the  goods  were  to  be  damaged  to  the  extent  of  $2368,  how  much 
would  each  company  pay  ? 

181.  What  is  the  interest  at  5%  on  £582  12s.  6d  from  January  14,  1908,  to 
September  22,  1909? 

182.  A note  for  $3246.27,  with  interest  at  4%,  dated  May  14,  1904,  has  the 
following  indorsements:  August  20,  1904,  $46.27  ; November  12,  1904,  $100; 
April  16,  1905,  $100;  July  23,  1905,  $100;  June  11,  1906,  $50  ; August  6,  1906, 
$25;  March  4,  1907,  $200;  June  17,  1907,  $500;  December  30, 1907,  $150;  April 
14,  1908,  $300.  How  much  is  due  August  1,  1908  ? United  States  rule. 

183.  If  it  takes  4 men  and  7 boys  2 days  to  do  a certain  piece  of  work, 
or  3 men  and  2 boys  3§|-  days  to  do  the  same  work,  how  long  would  it  take 
2 men  and  1 boy  to  do  it? 

181^.  Which  is  the  better  investment,  and  how  much  more  annual  income 
will  it  yield,  to  invest  $32000  in  5%  bonds  at  92|%,  brokerage  or  in  6% 
bonds  at  117f,  brokerage  £%,  reckoning  $1000  as  the  smallest  denomination 
purchased,  and  the  balance  to  be  deposited  in  a saving  fund  at  3%  interest? 

185.  An  invoice  of  merchandise  from  Amsterdam,  amounting  to  $12000 
guilders,  the  duty  on  which  was  40%,  and  freight  $432.50,  cost  altogether 
$7372.10;  at  what  rate  of  exchange  ivas  the  draft  in  settlement  purchased  ? 

186.  How  many  gallons  each  of  wines  worth  $1.15,  $1.25,  $1.40  and  $1.70 
a gallon,  respectively,  should  be  added  to  120  gallons  of  wine  worth  $1.45  a 
gallon  to  form  a mixture  worth  $1.35  a gallon? 

187.  A and  B form  a partnership,  agreeing  to  invest  equal  amounts  and 
share  equally  in  gains  or  losses,  and  further  agreeing  that  neither  shall  with- 
draw any  money  from  the  business  for  a period  of  five  years,  and  that  in 
case  the  yearly  statement  shall  show  a loss  at  the  end  of  any  year,  the}7  will 
make  additional  equal  investments  sufficient  to  make  up  the  loss.  At  the  end 
of  the  first  year  their  books  show  a net  gain  of  12J%  ; at  the  end  of  the 
second  year  a net  loss  of  8%  on  the  capital  at  the  beginning  of  this  year, 
which  they  make  good  by  additional  investments;  at  the  end  of  the  third  year, 
a net  gain  of  20%  on  this  year’s  capital;  at  the  end  of  the  fourth  year  a net 
loss  of  5%,  which  the}7  make  good;  at  the  end  of  the  fifth  year,  a net  gain 
of  18%.  Each  partner  now  has  $12744  to  his  credit.  What  amount  did  each 
invest  on  commencing  business,  and  what  was  his  average  profit  per  cent,  per 
annum  on  his  original  investment? 


GENERAL  REVIEW  PROBLEMS 


347 


188.  If  a cistern  16  feet  long,  8 feet  wide  and  12  feet  deep  can  be  filled  in 
8 hours  by  2 pipes,  each  1 inch  in  diameter,  how  long  will  it  take  3 pipes,  each 
T6g-  of  an  inch  in  diameter,  discharging  twice  as  fast,  to  fill  a cistern  18  feet  long, 
7 feet  wide  and  9 feet  deep  ? 

189.  What  is  the  balance  due  on  the  following  account  June  30,  1908, 
interest  at  6%  ? 


Dr.  B.  F.  Dodge  in  account  with  Kelly  & Co.  Or. 


T. 

E2 

i H 

02 

g 

02 

>< 

<5 

| 

53 

02 

> 

< 

g 

53 

P 

a 

H 

53 

e 

fc 

§ 

1908 

M 

< 

j 1908 

Hi 

< 

Jan. 

4 

Mdse.,  net 

*** 

** 

** 

2312118 

I Jan. 

29 

Cash 

*** 

** 

** 

2000 

Feb. 

11 

({ 

30  d. 

*** 

* 

** 

46750 

Feb. 

26 

i( 

*** 

* 

** 

350 

Mar. 

15 

(( 

60  d. 

*** 

** 

** 

712 

84 

Mar. 

1 

(6 

*** 

* 

** 

200 

Apr. 

21 

a 

30  d. 

** 

* 

** 

327 

48 

Apr. 

26 

U 

** 

* 

** 

500 

May 

19 

a 

net 

** 

* 

** 

674 

32 

May 

31 

it 

** 

* 

** 

500 

June 

3 

(( 

net 

** 

* 

** 

925 

75 

June 

30 

Interest  bal. 

** 

** 

« 

30 

Interest  bal. 

** 

** 

a 

30 

Balance 

** 

*** 

** 

** 

*** 

** 

** 

June 

30 

Balance 

** 

190.  The  specific  gravity  of  iron  is  7.84;  how  much  will  a hollow  globe  of 
iron  weigh,  whose  diameter  is  8 inches,  and  the  thickness  of  the  metal  f of  an 
inch  ? 

191.  How  many  feet  of  inch  lumber  will  be  required  for  the  floor  of  a plat- 
form 30  feet  long,  and  20  feet  wide  (from  front  to  back),  and  for  a roof  of  the 
same  material,  the  roof  to  project  one  foot  beyond  the  edge  of  the  platform  on 
all  sides,  and  to  be  supported  by  posts  16  feet  high  above  the  floor  in  front  and 
12  feet  above  it  at  the  back  ? What  will  the  lumber  cost,  at  $22  per  M? 

192.  A certain  excavation  consists  partly  of  clay  and  partly  of  rock.  If  it 
were  all  clay,  A could  do  the  work  in  20  days  ; if  it  were  all  rock,  he  could  do 
it  in  36  days.  If  it  were  all  rock,  B could  do  the  work  in  32  days  ; while  if  it  were 
all  clay  he  could  do  it  in  18  days.  A and  B work  together  on  the  job  and  finish 
it  in  lO-gy-g-  days  ; what  per  cent,  of  the  work  is  clay,  and  what  per  cent,  rock  ? 

193.  How  many  $1000  U.  S.  4s  at  122 J,  brokerage  §•%,  can  be  bought  with 
$32500,  and  what  will  be  the  balance  remaining  ? 

191/..  If  a man  purchases  $12000  U.  S.  4s  at  123§,  $5000  railroad  5s  at  102| 
and  $8000  railroad  6s  at  llOf,  through  a Philadelphia  stock  broker,  what  rate 
of  income  does  he  receive  on  the  total  investment? 

195.  How  many  thousand  ordinary  bricks  will  be  required  for  the  walls  of 
a round  tower  12  feet  in  diameter,  inside  measurement,  and  50  feet  high,  the 
walls  being  2 feet  thick  for  the  first  15  feet,  1 \ feet  thick  for  the  next  15  feet,  and 
1 foot  thick  for  the  remainder  of  the  height? 

196.  A man  purchased  a •§  interest  in  a factory,  and  afterwards  sold  18%  of 
his  share  at  a loss  of  7 % for  $1699.95 ; what  was  the  value  of  the  whole  factory 
at  the  rate  at  which  he  purchased  his  share  ? 


348 


GENERAL  REVIEW  PROBLEMS 


197.  8 lbs.  of  tea  and  5 lbs.  of  coffee  of  a certain  grade  are  worth  $6.10,  and 
3 lbs.  of  tea  and  7 lbs.  of  coffee  of  the  same  grade  are  worth  $3.62;  what  is  one 
pound  of  each  worth  ? 

198.  A grocer  sells  a blend  of  coffee  at  34  cents  a pound,  which  he  claims 
to  be  § Java  at  33  cents,  and  £ Mocha  at  36  cents;  the  Java  coffee  costs  him  30 
cents  a pound  and  the  Mocha  32  cents,  so  that  he  is  ostensibly  making  a profit 
ofl0-f%.  But  the  mixture  actually  contains  \ Rio  coffee,  which  costs  the 
grocer  only  18  cents  a pound  ; what  per  cent,  of  profit  does  he  really  make  ? 

199.  A commission  merchant  received  a consignment  of  cotton,  consisting 
of  260  bales  averaging  510  lbs.  to  the  bale,  which  he  sold,  and,  after  deducting 
his  commission  at  3%,  remitted  $10128.98  proceeds;  at  which  price  per  pound 
did  he  sell  the  cotton? 

SCO.  The  premium  paid  for  insuring  a stock  of  merchandise  at  2J%  for  £ of 
its  value  was  $689.87  ; what  was  the  value  of  the  merchandise  ? 

301.  A borrowed  from  B $1000  at  6%  interest,  agreeing  to  pay  $100  at  the 
end  of  each  year,  on  account  of  principal  and  interest,  until  the  whole  obligation 
was  canceled;  how  many  yearly  payments  did  he  make  and  what  was  the 
amount  of  his  last  payment  ? 

302.  Exchange  being  quoted  at  5.14,  what  will  be  the  cost  of  a draft  in 
settlement  of  an  importation  of  6 dozen  French  clocks,  invoiced  at  22.50  francs 
each;  for  how  much  apiece  must  they  be  sold  in  Philadelphia  to  gain  30%, 
if  the  duty  is  25%  and  other  charges  in  Philadelphia  amount  to  $37.40  ? 

203.  If  14  men  can  build  a wall  60  feet  long,  5 feet  high.  If  feet  thick  in 
20  days,  how  many  men  will  be  required  to  build  a wall  340  feet  long,  44  feet 
high,  24  feet  thick  in  50  days  ? 

201^.  If  a lot  of  goods  is  bought  for  $327.42,  and  f of  it  is  sold  at  a gain  of 
30%,  and  £ of  the  remainder  proves  unsalable,  how  should  the  other  f of  the 
remainder  be  marked  so  that  a discount  of  10%  may  be  allowed  and  yet  enough 
gained  to  net  25%  profit  on  the  whole  purchase  ? 

205.  A piece  of  ground  in  the  form  of  a circle  has  a diameter  of  16£  rods ; 
what  will  it  cost,  at  20  cents  a cubic  yard,  to  dig  a ditch  around  it,  outside,  24 
feet  wide,  and  3J  feet  deep  ? 

206.  What  per  cent,  is  gained  by  buying  coal  at  $3.75  a long  ton  and  selling 
at  $5  a short  ton  ? 

207.  A boy’s  father  deposits  $300  in  a savings  bank  wrhen  the  boy  is  16 
years  old.  If  this  bank  allows  1 % interest  quarterly,  and  balances  its  books 
quarterly  but  reckons  no  interest  on  fractions  of  a dollar,  how  much  will  there 
be  to  his  credit  when  he  reaches  the  age  of  21  ? 

208.  A commission  merchant  in  New  York  sells  for  me  1200  baskets  of 
potatoes  at  48  cents  a basket;  he  pays  freight  and  other  charges  $112.80; 
deducts  his  commission  of  3%  ; and  invests  the  proceeds  in  flour,  charging  2% 
for  purchasing.  If  he  buys  100  barrels  of  flour,  and  remits  a balance  of  $7. IS, 
at  what  price  does  he  purchase  ? 


GENERAL  REVIEW  PROBLEMS 


349 


209.  If  an  insurance  company  paid  $3920,  under  its  policy  of  $20000  with 
u average  clause,”  on  a loss  of  $6272,  what  was  the  value  of  the  property 
insured  ? 

210.  A works  3 days  on  a certain  piece  of  work,  then  B works  4 days,  after 
which  A finishes  the  work.  If  it  takes  A alone  21  days  to  do  the  work,  or  A and 
B together  12  days,  how  long  will  it  take  A to  finish  it  ? 

211.  Sold  $15000  U.  S.  4s  at  121f,  and  invested  the  proceeds  in  railroad 
stock  at  48f,  which  was  afterwards  sold  at  52J.  What  was  the  gain  on  the  stock, 
if  Philadelphia  rates  of  brokerage  were  paid? 

'212.  The  assessed  valuation  of  the  property  in  a certain  town  is  $2347896. 
The  amount  to  be  raised  by  taxation  is  $48394.50.  What  is  the  tax  rate  ? What 
is  the  tax  on  $100  ? 

213.  What  is  the  duty  at  $1  per  pound  on  an  importation  of  partly  manu- 
factured silk  from  France,  weighing  874  pounds  and  invoiced  at  9098.45  francs, 
with  the  proviso  that  “ in  no  case  shall  the  duty  be  less  than  fifty  per  centum 
ad  valorem  of  the  invoiced  cost”  of  the  goods? 

21/, l.  What  would  be  the  cost,  exchange  at  5.14,  of  a draft  to  pay  for  the 
importation  in  Problem  213? 

215.  How  many  pounds  each  of  teas  worth  42,  48,  54,  60  and  75  cents  per 
pound,  respectively,  should  be  taken  to  form  a mixture  of  80  lbs.  at  55  cents? 

216.  If  it  takes  22  men,  working  8 hours  a day  and  5 days  a week,  4 weeks  to 
do  a certain  piece  of  work,  how  many  weeks  will  it  take  40  boys,  working  10  hours 
a day  and  4 days  a week,  to  do  half  as  much,  9 boys  doing  as  much  as  7 men? 

217.  Find  the  contents  of  a triangular  prism  4 feet  high,  the  sides  of  the 
base  being  14,  19  and  26  inches,  respectively. 

218.  Bought  a car  load  of  wheat,  34263  lbs.,  at  83  cents  a bushel,  and  a car 
load  of  corn,  32796  lbs.,  at  56  cents  a bushel.  Sold  the  wheat  at  a profit  of  20% 
and  the  corn  at  a profit  of  16§ %.  What  was  the  total  gain? 

219.  A,  B and  C form  a partnership,  agreeing  to  divide  gains  or  losses 
equally,  under  the  following  conditions : B is  to  be  general  manager  at  a salary 
of  $2000  per  annum,  and  C bookkeeper  at  $1500  per  annum,  their  salaries  to  be 
credited  quarterly  at  the  end  of  each  quarter.  Interest  is  to  be  charged  and 
allowed  at  6%  per  annum.  A is  to  furnish  J of  the  capital,  and  B and  C each  \. 
A invests  January  1,  $12000 ; B $6000,  and  C $6000.  A withdraws  March  1, 
$675;  June  1,  $500,  and  September  1,  $900.  B withdraws  February  1,  $300; 
May  1,  $500  ; August  1,  $400,  and  October  1,  $375.  C withdraws  April  1, 
$500;  July  1,  $500;  September  1,  $400,  and  November  1,  $400.  A invests 
$2000  additional  on  July  1.  The  net  gain  for  the  year,  exclusive  of  salary  and 
interest  items,  is  $4760.83  ; what  is  the  balance  of  each  partner’s  account  ? 

220.  A pile  of  coal  in  the  corner  of  a cellar  reaches  within  1 foot  of  the 
flooring  above,  which  is  8 feet  above  the  cellar  floor,  and  extends  10  feet  along 
each  wall  at  the  bottom  of  the  pile,  the  pile  having  a plane  slope  (so  that  it 
forms  J of  a pyramid).  How  much  coal, dong  tons,  is  there  in  the  pile,  reckoning 
80  lbs.  to  the  bushel  of  2150.4  cu.  in.  ? 


350 


GENERAL  REVIEW  PROBLEMS 


221.  What  amount  is  due  on  the  following  account  January  1, 1908,  interest 
at  6%  ? 

Dr.  Ambrose  C.  Headley  Or. 


1908 

1908 

Jan. 

4 

Mdse. 

net 

757 

14 

Mar. 

11 

Cash 

1500 

00 

Feb. 

10 

u 

30  days 

1213 

19 

Apr. 

15 

(( 

800 

00 

Mar. 

18 

u 

60  days 

472 

18 

May 

1 

Note,  30  days 

215 

00 

Apr. 

15 

u 

30  days 

327 

65 

June 

15 

Cash 

500 

00 

June 

9 

a 

net 

782 

80 

July 

1 

U 

300 

00 

July 

22 

u 

net 

485 

25 

Aug. 

10 

U 

200 

00 

Aug. 

19 

u 

60  days 

379 

92 

Sept. 

1 

Note,  30  days 

300 

00 

Sept. 

30 

(( 

3 mos. 

915 

20 

Oct. 

15 

Cash 

750 

00 

Nov. 

3 

u 

30  days 

455 

70 

Dec. 

15 

u 

60  days 

1114 

22 

222.  A certain  country  paid  4J%  interest  on  its  debt.  A war  increased  the 
debt  25%.  During  peace  which  followed,  the  debt  was  diminished  $25000000, 
and  the  rate  was  reduced  to  4%.  The  annual  interest  was  then  the  same  as  at 
first.  What  was  the  indebtedness  of  the  county  before  the  war? 

223.  A piece  of  gold,  alloyed  with  silver,  is  14  carats  fine  and  weighs  72 
pennyweights.  How  much  gold  must  be  added  to  make  it  18  carats  fine? 

22 If..  A man  drew  out  50%  of  the  amount  he  had  to  his  credit  in  bank,  and 
gave  16§  % of  the  money  withdrawn  for  a house,  for  wThich  he  paid  $4000.  He 
had  the  house  insured  for  $3000  at  f %.  He  invested  the  balance  of  the  money 
withdrawn  from  bank  in  wheat,  which  he  sold  for  cash,  J at  an  advance  of  30%, 
| at  an  advance  of  18%,  and  | at  a loss  of  71%.  The  house  was  destroyed  by 
fire  and  the  insurance  company  paid  the  amount  of  their  policy,  which  the  man 
deposited  in  bank,  together  with  all  his  other  receipts.  How  much  had  he  in 
bank  originally,  what  was  the  final  balance  of  his  bank  account,  and  what  was 
his  per  cent,  of  loss  or  gain  ? 

225.  A Philadelphia  commission  merchant  received  a consignment  of  wheat 
to  sell,  the  total  wreight  of  which  wras  423632  lbs.  He  paid  freight  and  other 
charges  amounting  to  $1080.50;  sold  at  98  cents  per  bushel;  charged  21% 
commission;  and  remitted  the  proceeds  by  bank  draft  on  Chicago,  purchased 
at  |%  premium.  How  much  did  the  consignor  receive? 

226.  A and  B have  15  acres  to  plow.  At  the  end  of  1J  days  A leaves,  and 
B finishes  in  3f  days.  If  B had  left  instead  of  A,  it  would  have  taken  A 21  days 
to  finish.  How  long  would  it  take  each  to  plow  the  field  alone? 

227.  If  I buy  4%  bonds  maturing  in  28  years,  at  125J,  how  much  per  cent, 
greater  or  less  is  my  annual  income  than  if  I buy  4%  bonds  maturing  in  10 
years,  at  112?  (Brokerage  \ % in  each  case.) 

228.  A walks  at  the  rate  of  3f  miles  an  hour  and  starts  IS  minutes  before  B. 
At  what  rate  per  hour  must  B walk  to  overtake  A at  the  ninth  mile-stone? 


GENERAL  REVIEW  PROBLEMS 


351 


229.  Divide  175  into  four  such  parts  that  the  first  plus  2,  the  second  minus 
3,  the  third  multiplied  by  4,  and  the  fourth  divided  by  5,  shall  be  equal  to  each 
other. 

230.  Find  the  equated  date  for  payment  of  the  balance  of  the  following 
account : 

Dr.  Harolb  S.  Warner  Or. 


1908 

Apr. 

7 

Mdse.,  60  days 

2314 

75; 

1908 

June 

9 

Cash 

2000 

00 

June 

11 

“ 30  days 

6218 

90 

July 

15 

U 

5000 

00 

Aug. 

19 

“ net 

572 

20 

Aug. 

27 

Note,  30  days 

2000 

00 

Oct. 

7 

“ 30  days 

376 

19 

I Nov. 

12 

Cash 

250 

00 

Dec. 

2 

“ net 

1042 

18 

231.  The  specific  gravity  of  silver  being  10.4,  what  will  a spherical  ball  of 
silver  weigh  whose  diameter  is  7 inches? 

232.  A merchant  added  $1700  to  his  capital  the  first  year;  during  the 
second  year  he  further  increased  it  by  a sum  equal  to  10%  of  his  original  capi- 
tal ; during  the  third  year  he  lost  40%  of  what  he  had  at  the  end  of  the  second 
year,  and  found  that  he  then  had  just  wdiat  he  began  with.  What  was  his 
original  capital  ? 

233.  If  a draft  on  London  for  £2348  13s.  Id.  cost  $11482.11,  what  was  the 
rate  of  exchange? 

234..  A man  borrowed  enough  money  at  6 per  cent,  to  pay  for  a house  and 
also  for  repairs  amounting  to  2%  of  the  purchase  money.  The  house  was  vacant 
for  a year,  and  during  that  time  he  had  to  pay  $34.60  taxes.  At  the  end  of  the 
year  he  sold  the  house  for  $4600  and  found  his  net  loss  to  be  6§  % of  the  pur- 
chase price.  What  did  the  house  cost? 

235.  How  many  square  yards  are  there  in  the  surface  of  the  sidewTalk  around 
a city  square  244  feet  X 480  feet  on  the  building  line,  if  the  sidewalk  is  18  feet 
wide  ? 

236.  A and  B have  equal  incomes.  A’s  expenses  are  16f  % more  than  his 
income,  while  B lives  on  75%  of  his.  At  the  end  of  three  years,  B lends  A 
enough  money  to  pay  his  debts,  and  has  $150  left.  What  is  the  income  of  each  ? 

237.  What  per  cent,  is  gained  by  buying  silk  at  4.95  francs  a meter,  and 
selling  it  at  $1.87J  a yard,  if  the  duty  is  50%  and  exchange  5.14J? 

238.  How  many  gallons  of  water  will  a cylindrical  boiler  hold,  12  feet  long 
and  5 feet  diameter,  inside  measurement,  if  there  are  160  flues,  each  3 inches 
outside  diameter,  passing  through  it  lengthwise? 

239.  A merchant  increased  his  capital  the  first  year  by  10%  of  itself ; the 
second  year  he  gained  20%  ; the  third  year  he  lost  25%.  He  then  had  $100  less 
than  at  first.  What  was  his  original  capital  ? 

211.0.  At  his  death,  A’s  property  was  valued  at  $42360 ; $27360  being  in  real 
estate  and  the  remainder  in  personal  property.  His  heirs  were  the  widow,  three 
sons  and  two  daughters.  The  will  provided  that  the  widow  should  receive  ^ of 
the  real  estate  and  20%  of  the  personal  property  and  that  the  remainder  should 


352 


GENERAL  REVIEW  PROBLEMS 


be  divided  among  the  sons  and  daughters,  each  daughter  receiving  25%  less 
than  a son.  What  was  the  share  of  each? 

241.  What  per  cent,  of  a kilogram  is  a pound? 

21$.  In  the  following  account  find  the  balance  due  February  1,  1909,  reck- 
oning by  “ daily  balances,”  charging  6%  interest  on  debit  balances  and  allowing 
4%  interest  on  credit  balances. 

Dr.  R.  J.  Melrose  Or. 


— 

E- 

m 

— 

E- 

m 

DAYS 

g 

W 

H 

Eh 

P 

O 

DAYS 

® H 

« g 

Eh  a 

1909 

Z 

5 

1909 

£ 

Jan. 

4 

500  Penna. 

26187 

50 

Jan. 

1 

Balance 

42128 

17 

U 

7 

1500  Readg. 

18177 

50 

CC 

4 

Interest 

** 

CC 

12 

400  Erie  pfd. 

28900 

cC 

7 

CC 

* 

CC 

12 

Interest 

* 

* 

'fc  ¥ 

CC 

18 

1500  Readg. 

18875 

CC 

18 

cc 

* 

% 

** 

cc 

22 

400  Erie  pfd. 

29500 

22 

* 

jj; 

** 

CC 

26 

500  Penna. 

26750 

Feb. 

1 

Interest  bal. 

** 

CC 

26 

Interest 

* 

** 

cc 

1 

Balance 

^ >K  ^ 5^ 

>j<  1 

Feb. 

1 

»C  * 

** 

** 

CC 

1 

Interest  bal. 

JjC  JjC  jfc  ^ 

** 

** 

** 

1909 

Feb. 

1 

Balance 

24S.  Sent  12000  bushels  of  wheat  to  my  agent,  which  he  sold  at  58  cents  per 
bushel.  He  paid  expenses  $214.52  and  deducted  his  commission  of  21%.  He 
then  invested  the  proceeds  in  sugar  at  5|  cents  per  pound,  commission  for  buy- 
ing 2%.  How  many  pounds  did  he  purchase? 

244-  A merchant’s  sales  increased  the  second  year  20%  over  the  first  year; 
the  third  year  25%  over  the  second,  and  the  fourth  year  40%  over  the  third. 
During  the  four  years  he  sold  $131250  worth  of  goods.  What  was  the  amount 
of  his  sales  the  first  year? 

245.  James  Farley  has  $352  in  bank  and  has  a note  for  $850  coming  due 
to-day  which  has  been  made  payable  at  his  bank.  For  how  much  must  he  give 
his  note  at  four  months,  dated  to-day,  that  when  discounted  the  bank  may  pay 
his  $850  note  and  leave  a balance  to  his  credit  at  $55.25  ? 

246.  How  much  must  I invest  in  U.  S.  4s  at  122J,  brokerage  |%,to  secure 
a quarterly  dividend  of  $460? 

247.  When  is  the  following  account  due  bjr  equation,  and  what  amount  is 
due  June  1,  1909  ? 

Dr.  L.  T.  Spencer  Or. 


1908 

Jan. 

1 

Balance 

694 

87 

1908 

Feb. 

20 

Cash 

550 

00 

CC 

28 

Mdse.,  30  days 

1400 

00 

a 

28 

Draft,  20  days 

700 

00 

Mar. 

20 

“ net 

510 

40 

Mar. 

31 

Cash 

S45 

50 

Apr. 

11 

“ 10  days 

383 

S3 

GENERAL  REVIEW  PROBLEMS 


353 


248.  What  amount  will  be  required  on  September  16,  1909,  to  pay  the 
balance  due  on  a note  for  $720,  with  interest  at  4%,  dated  September  17,  1908, 
upon  which  the  following  payments  were  made  : October  8,  1908,  $125  ; January 
15,  1909,  $15 ; February  23,  1909,  $333  ; March  1,  1909,  $225  ? U.  S.  Rule. 

24-9.  A merchant  sold  a lot  of  goods  at  124%  off  list  and  another  lot  of  the 
same  value  at  20%  and  10%  off  list.  He  then  allowed,  on  the  'whole  invoice,  a 
discount  of  3%  for  cash.  What  was  the  value  of  each  lot  of  goods  at  list  price, 
if  the  cash  paid  was  $3841.20? 

250.  If  the  interest  on  $510  at  6%  for  4 years  and  9 months  is  $145.35,  what 
will  be  the  interest  on  $1350  for  the  same  time  at  the  same  rate  ? 

251.  An  importation  of  1312  meters  of  silk  from  Lyons,  France,  was  invoiced 
at  3.5  francs  per  meter.  The  importer  purchased  a bill  of  exchange  to  pay  for  it 
at  5.15.  The  duty  was  50%,  and  other  expenses  $42.50.  At  what  price  per  yard 
must  the  silk  be  sold  to  gain  25  % ? 

252.  Suppose  a globe  to  be  inclosed  in  a cylinder  that  will  exactly  contain 
it,  and  the  cylinder  is  to  be  inclosed  in  a cube  that  will  exactly  contain  it ; what 
decimal  part  of  the  volume  of  the  cube  is  the  volume  of  the  cylinder,  what  part 
of  the  volume  of  the  cylinder  is  the  volume  of  the  sphere,  and  what  part  of  the 
volume  of  the  cube  is  the  volume  of  the  sphere? 

253.  If  the  diameter  of  the  base  of  a cone  and  the  diameter  of  a hemisphere 
are  equal,  and  the  altitude  of  the  cone  is  equal  to  the  radius  of  the  hemisphere, 
what  is  the  ratio  of  the  volume  of  the  cone  to  the  volume  of  the  hemisphere? 

254 . Find  the  cost  of  a sheet-iron  smoke-stack  40  feet  high  and  2 feet  in 
diameter  at  15  cents  per  square  foot. 

255.  Merchandise  weighing  8743  lbs.  is  transported  1472  miles  at  a through 
rate  of  $3.24  per  100  pounds.  The  first  road  carried  it  329  miles,  the  second  658 
miles,  the  third  219  miles,  and  the  fourth  forwarded  it  to  its  destination.  How 
shall  the  freight  be  apportioned,  the  fourth  road  receiving  a 2-cent  terminal  ? 

256.  An  importer  received  an  invoice  of  350.5  meters  of  silk  at  5.85  francs 
per  meter.  He  paid  duty  at  50%,  and  other  charges  $25.60.  Exchange  for  his 
remittance  cost  him  at  the  rate  of  5.13J.  What  price  per  yard  must  he  receive 
for  the  silk  in  order  to  gain  25  % ? 

257.  Bought  goods  at  five  dollars  a gross,  20%  off,  and  sold  them  at  fifty 
cents  a dozen,  20%  and  10%  off.  Is  there  a profit  or  a loss,  and  what  is  the  rate 
per  cent.  ? 

258.  When  is  the  balance  of  the  following  account  due  by  equation  ? 


Dr.  Francis  E.  Hoyt  Cr. 


" 1908 

1908 

May 

8 Mdse.,  30  days 

0 

0 

CO 

oo1 

May  20 

Cash 

150 

00 

June 

20  “ 60  days 

463 

29; 

July  ; 1 

Note,  1 mo. 

412 

50 

Aug. 

28  “ net 

CO 

- 1 

-1 

89: 

Sept.  | 2 

Cash 

200 

00 

354 


GENERAL  REVIEW  PROBLEMS 


259.  At  63  cents  a square  foot,  what  is  the  cost  of  a building  lot  having  two 
parallel  sides,  respectively  140  feet  and  116  feet  long,  and  83  feet  apart? 

260.  If  a draft  on  Philadelphia,  payable  60  days  after  date,  was  bought  in 
St.  Louis  for  $3723.29  at  \ % premium,  what  was  its  face? 

261.  Find  the  height  and  area  of  a triangle  whose  base  is  42  feet  long,  the 
other  sides  being  each  35  feet  long. 

Note. — Observe  that  this  triaDgle  is  equal  to  two  rigid-angled,  triangles,  each  having  a hypot- 
enuse of  35  feet  and  a base  of  21  feet. 

262.  A man  sold  two  properties  for  $8522.50  ; on  the  first  he  lost  124%,  and 
on  the  second  he  gained  30%.  What  did  he  gain  on  the  whole  transaction,  if 
f of  what  he  gave  for  the  first  property  was  equal  to  J of  what  he  gave  for  the 
second  ? 

263.  Having  sent  a New  Orleans  agent  $1836.46  to  be  invested  in  sugar, 
after  allowing  3%  on  the  investment  for  his  commission,  I received  32400 
pounds  of  sugar.  At  what  price  per  pound  did  the  agent  purchase? 

264..  A buys  a bill  of  goods  amounting  to  $6776.40,  20%  and  10%  off  on  the 
following  terms : “Four  months,  or  less  3%  for  cash.”  He  accepts  the  latter, 
and  borrows  the  money  at  6%  to  pay  the  bill.  How  much  does  he  gain? 

265.  What  is  the  weight  of  a pint  of  alcohol,  if  its  specific  gravity  is  .792? 

266.  A draft  on  Philadelphia  for  $9375.15  was  purchased  in  Chicago  at  \ % 
discount.  If  the  draft  cost  $9300.16,  at  how  many  days  after  date  was  it 
drawn  ? 

267.  A and  B together  can  do  a certain  piece  of  work  in  11J  days;  A and 
C together  can  do  it  in  10|-  days ; B and  C together  can  do  it  in  14-f  days.  If  A 
and  B work  together  for  2 days,  and  then  A continues  alone  for  3 days,  how  long 
would  it  take  C to  finish  the  work  alone  ? 

268.  A commission  merchant  in  Chicago  sells  for  me  12  bales  brown  sheet- 
ing, each  bail  containing  800  yards,  at  7 cents  per  yard  ; pays  transportation 
and  other  charges  amounting  to  $72;  and  invests  the  proceeds  in  flour.  If  he 
charges  24%  for  selling  and  for  purchasing,  and  sends  me  120  barrels  of 
flour,  at  what  price  does  he  buy  ? 

269.  What  is  the  balance  due  on  the  following  account  November  1,  1908, 
interest  at  6 % ? 


Dr.  Walter  G.  Farnsworth  Oi'. 


1908 

1908 

Mar. 

27 

Mdse.,  30  days 

243748 

Apr.  7 

Cash 

2500 

U 

30 

“ net 

51760 

June  1 

(( 

3000 

May 

12 

“ 30  days 

3224  71 

July  1 

Draft,  10  days 

2500 

28 

“ net 

327  14 

Aug.  1 9 

Cash 

4000 

June 

10 

“ 30  days 

1895  36 

Sept.  17 

U 

1800 

July 

21 

“ 30  “ 

524307 

Sept. 

8 

“ 30  “ 

678  25 

Oct. 

15 

“ 30  “ 

132427 

GENERAL  REVIEW  PROBLEMS 


355 


370.  A merchant  sold  a bill  of  goods  at  20%,  10%  and  5%  off  list  price,  and 
allowed  a discount  of  3%  for  cash.  What  was  the  list  price,  if  the  cash  paid 
was  $3481.20? 

371.  Since  the  volumes  of  similar  solids  are  to  each  other  as  the  cubes  of 
their  corresponding  dimensions,  how  many  steel  balls  \ of  an  inch  in  diameter 
will  weigh  as  much  as  one  2 inches  in  diameter? 

373.  A column  10  inches  in  diameter  and  18  feet  high  is  to  be  gilded  with 
a spiral  stripe  2J  inches  wide,  winding  around  it  so  that  the  turns  are  7 inches 
apart.  What  will  it  cost  for  the  gold  leaf  needed,  at  $3.20  per  book  of  100 
leaves  3X4  inches,  estimating  that  one-sixth  of  the  material  will  be  wasted  ? 

373.  An  exporter  shipped  400  cases  of  canned  goods  to  Liverpool,  invoiced 
at  14s.  10 d.  per  case  and  drew  on  the  consignees  for  the  amount  of  the  invoice, 
selling  the  draft  at  4.87.  He  shipped  a similar  lot,  400  cases,  to  Havre,  invoiced 
at  18.70  francs  per  case,  and  sold  his  draft  on  consignees  at  5.15.  For  which 
draft  did  he  receive  the  more  money,  and  how  much  was  the  difference? 

37 It..  How  many  cubic  inches  of  lead  will  weigh  as  much  as  12  bushels  of 
wheat?  (The  specific  gravity  of  lead  is  11.35.) 

375.  A note  for  $327.50,  with  interest  at  5%,  dated  March  21,  1908,  has  the 
following  indorsements:  June  1,  1908,  $25;  July  15,  1908,  $10;  September  3, 
1908,  $50;  October  20,  1908,  $100  ; November  10,  1908,  $20;  December  8,  1908, 
$30.  How  much  is  due  February  17,  1909? 

376.  If  I pay  $9.48  interest  on  $555  for  123  days,  what  is  the  rate  per  cent.  ? 

377.  Loaned  $2400  at  6%  simple  interest  until  it  amounted  to  $2998.80. 
For  what  time  was  the  loan  made? 

378.  Two  clerks  have  2000  circulars  to  address.  One  .quits  at  the  end  of  4 
hours,  and  it  takes  the  other  10  hours  to  finish.  If  the  one  who  left  had 
remained  2 hours  longer,  the  other  could  have  finished  in  5 hours.  How  many 
can  each  address  in  1 hour? 

379.  Bought  in  Manchester,  England,  8 gross  of  razors  at  £3  13s.  6d.  a dozen, 
less  10%  and  5%  ; what  sum  in  United  States  money  is  equivalent  to  the  net 
amount  of  the  invoice? 

380.  A commission  merchant  sold  83748  lbs.  of  cotton  at  7f  cents  a pound, 
paid  transportation  $368.72,  and  cartage  $8,  and  charged  3%  commission. 
What  per  cent,  of  profit  did  the  consignor  make,  if  the  cotton  cost  him  $5600  ? 

381.  A shipment  of  300  cases  of  merchandise,  valued  at  $87.50  a case,  was 
insured  for  $20000,  the  policy  containing  the  “ average  clause.”  In  consequence 
of  a fire  at  the  railroad  depot,  42  cases  of  the  goods  were  partly  damaged  ; and 
the  proceeds  of  the  damaged  lot,  when  sold  by  auction,  amounted  to  $843.50. 
What  amount  was  recovered  on  the  policy  ? 

383.  A can  do  f of  a certain  piece  of  work  in  4 days ; B can  do  -f-  of  it  in  3 
days  ; C can  do  f of  it  in  7 days.  If  A begins  the  work  alone  and  works  for  3 
days,  and  then  B and  C relieve  him,  working  together  for  2f  days,  how  long 
would  it  take  A and  C together  to  finish  the  work  ? 


356 


GENERAL  REVIEW  PROBLEMS 


283.  Received  $1436.84  on  September  10,  1908,  as  the  amount  of  a loan  at 
4J%,  made  June  14,  1907  ; what  was  the  principal? 

284..  A speculator  deposited  $5000  witli  his  broker,  who  10  days  later  pur- 
chased for  him,  500  shares  of  stock  at  92§-  (brokerage  \%).  The  stock  was  sold 
23  days  afterward  at  94f-  (brokerage  \%),  and  settlement  made  the  same  day. 
How  much  did  the  speculator  receive  from  the  broker  (interest  at  6%)? 

285.  The  tax  on  a certain  property,  at  $18.93  on  $1000,  amounts  to  $460.32  ; 
what  is  the  assessed  valuation  of  the  property  ? 

286.  What  is  the  duty  on  680  cubic  feet  of  marble,  invoiced  at  21465  lire, 
at  $1  per  cubic  foot  and  25%  ad  valorem? 

287.  A draft  on  San  Francisco  for  $6850.75,  payable  60  days  after  date,  was 
purchased  at  f%  discount;  how  much  did  it  cost,  money  being  worth  6%  ? 

288.  How  much  tea  at  22  cents,  28  cents  and  50  cents  a pound  must  be 
mixed  with  45  pounds  at  64  cents  a pound,  so  that  the  whole  may  be  sold  at  48 
cents  a pound  and  produce  a gain  of  20%  ? 

289.  X and  Q are  partners,  sharing  gains  or  losses  in  proportion  to  average 
net  investment.  On  January  1 X invests  $18000,  and  Q invests  $15000.  X 
draws  out  $350  at  the  end  of  each  month  during  the  year,  and  Q draws  out  S300 
on  the  15th  of  each  month.  At  the  end  of  the  year  they  have  a net  gain  of 
$4350  to  divide ; how  much  does  each  receive  ? 

290.  What  is  the  cash  balance  of  the  following  account  on  January  1, 1909  ? 


Dr.  Calvin  Thompson  Cr. 


1908  | 

1908 

Jan.  j 5 

Mdse.,  60  days 

389 

26 

Apr.  1 Cash 

600 

Feb.  4 

U 

30  “ 

414 

70 

June , 1 “ 

500 

Mar.  12 

(( 

90  “ 

853 

25 

Aug.  2 “ 

500 

Apr.  10 

u 

net 

721 

19 

Oct.  1 “ 

1000 

June  17 

a 

90  days 

436 

20 

Dec.  15  “ 

1500 

Aug.  12 

u 

30  “ 

525 

00 

Sept.  9 

u 

net 

324 

50 

Nov.  10 

u 

90  days 

622 

80 

Dec.  122 

“ 

60  “ 

875 

70 

291.  An  ice  company  has  its  men  at  work  on  a lake  cutting  10-inch  ice. 
If  ice  weighs  58  lbs.  to  the  cubic  foot,  and  20300  lbs.  can  be  loaded  in  a car,  bow 
long  a strip  100  feet  wide  will  the  men  cut  to  fill  40  cars  ? 

292.  In  digging  a well  4 feet  in  diameter,  37  cubic  yards  ot  earth  were  taken 
out;  what  was  the  depth  of  the  well  ? 

293.  At  what  rate  will  $2468,  in  1 yr.  1 mo.  13  da.  amount  to  $2744.28? 
294 ■ The  net  proceeds  of  a sale  were  $3214.78  ; the  charges  $212.50,  and  4% 

commission.  What  were  the  gross  proceeds  ? 

295.  Goods  that  were  purchased  for  $2319.75,  less  30%,  10%  and  4%,  were 
marked  40%  above  net  cost  and  sold  at  20%,  124%  and  3%  off  the  marked 
price  ; how  much  was  the  loss  ? 


GENERAL  REVIEW  PROBLEMS 


357 


296.  How  many  reams  of  paper,  in  sheets  24X38  inches,  will  be  required  to 
print  12000  copies  of  a book  of  288  pages,  each  page  measuring  6X9|  inches 
(untrimmed) ; and  how  much  will  this  amount  of  paper  cost,  at  6 cents  per 
pound,  if  it  weighs  60  pounds  to  the  ream  ? 

297.  What  per  cent,  of  the  volume  of  a pound  of  lead  is  the  volume  of  a 
pound  of  gold  ? 

Note. — This  is  a problem  in  compound  proportion.  The  specific  gravity  of  lead  is  11.35  ; that  of 
gold  19-26.  A pound  of  gold  weighs  5760  grains  ; a pound  of  lead,  7000  grains. 

298.  The  polar  diameter  of  the  earth  is  7899  miles;  the  equatorial  diameter 
is  7925J  miles.  How  many  cubic  miles  less  is  the  volume  of  the  earth,  than  if 
it  were  a perfect  sphere  8000  miles  in  diameter  ? 

299.  A hexagonal  prism  is  5 feet  6 inches  in  height ; the  distance  from  the 
center  of  the  base  to  the  center  of  each  side  of  the  base  is  17.32  inches ; and  the 
length  of  each  side  of  the  base  is  20  inches.  Give  the  contents  of  the  prism  in 
cubic  feet. 

300.  A,  B and  C do  a piece  of  work.  The  portions  done  by  A and  B 
together  equal  ^ of  the  whole  work ; the  portions  done  by  B and  C together 
equal  T7T  of  the  whole.  What  part  of  the  work  was  done  by  B? 

301.  The  sides  of  two  square  fields  are  to  each  other  as  3 to  5,  and  the  total 
area  of  the  two  fields  is  3f  acres ; what  is  the  length  of  a side  of  each  field,  in 
yards  ? 

302.  If  the  driving  wheels  of  a locomotive  are  4 feet  4 inches  in  diameter,  at 
the  rate  of  how  many  miles  per  hour  is  the  locomotive  running  when  the  driving 
wheels  are  making  312  revolutions  per  minute? 

303.  A,  B and  C enter  into  partnership  on  March  1,  1907,  A furnishing  J of 
the  capital  and  each  of  the  others  J;  they  agree  to  share  gains  or  losses  in  this 
proportion,  each  partner  being  charged  6%  interest  on  any  sums  he  may  with- 
draw during  the  year.  A invests  $10000,  B and  C each  $5000.  A withdraws 
during  the  year  as  follows:  May  6,  1907,  $212.50;  July  23,  1907,  $67.20; 
October  12,  1907,  $132.70;  January  20,  1908,  $74.75;  February  3,  1908,  $65.  B 
withdraws : April  16,  1907,  $84.30 ; July  2,  1907,  $105 ; September  28,  1907, 
$38.75  ; December  24,  1907,  $165.80.  C withdraws:  June  10,  1907,  $90;  August 
31,  1907,  $123.40;  November  27,  1907,  $218.50.  On  March  11,  1908,  before  the 
interest  on  withdrawals  has  been  computed,  the  books  of  the  firm  show  a net  loss 
of  $1837.92  ; what  is  the  balance  of  each  partner’s  account  after  the  books  have 
been  closed  for  the  year? 

30 J.  A commission  merchant  in  Philadelphia  sold  84723  lbs.  of  corn  at  48|- 
cents  a bushel,  paying  $32.40  charges,  and  receiving  a commission  of  2J%.  He 
then  purchased  merchandise  according  to  the  consignor’s  directions,  the  invoice 
amounting  to  $324.30,  on  which  he  received  a commission  of  The  balance 

due  the  consignor  was  remitted  by  draft  at  30  days  after  date,  purchased  at  f % 
discount ; what  was  the  face  of  the  draft  ? 


358 


GENKRAL  REVIEW  PROBLEMS 


305.  A room  is  27  ft.  3 in.  long,  23  ft.  5 in.  wide.  If  carpeted  with  a border 
22  in.  wide  an  allowance  of  16  in.  must  be  made  on  each  strip  for  matching, 
strips  laid  lengthwise,  carpet  27  in.  wide;  but  if  no  border  is  used  an  allowance 
of  1 ft.  8 in.  mast  be  made  on  each  strip.  What  will  it  cost  to  carpet  the  room 
with  a border  and  without  a border,  carpet  and  border  costing  $1.20  a j^ard?  In 
each  case  there  is  a lining  1 yard  wide  at  10  cents  a yard. 

306.  What  will  be  the  cost  of  carpet  f of  a yard  wide,  and  selling  at  $2.25  per 
yard,  and  of  lining  f of  a yard  wide,  costing  30  cents  per  yard,  to  cover  a room 
24  feet  long  and  20  feet  wide;  the  strips  of  carpet  are  laid  lengthwise,  and  there 
is  a waste  of  9 inches  to  each  strip  of  carpet  in  matching,  also  an  allowance  of 
10%  in  width  and  6%  in  length  for  shrinkage  of  the  lining? 

307.  The  assessed  valuation  of  the  real  estate  of  a township  is  $910887,  and 
of  the  personal  property,  $521073  ; it  has  3564  inhabitants,  subject  to  a poll-tax 
of  $1.25.  The  year’s  expenses,  not  including  the  collector’s  commission  of  2%, 
are : for  schools,  $4400  ; interest,  $3850 ; highways,  $4560  ; salaries,  $3150  ; and  for 
contingent  expenses,  $8675.  If  $585  is  in  the  treasury  and  two  hotels  each  pay 
a license  of  $250  into  the  treasury  and  the  revenue  from  fairs  amounts  to  $2500, 
what  tax  must  be  levied  on  a dollar,  to  meet  expenses,  and  provide  a sinking 
fund  of  $5000?  What  would  be  A’s  tax  whose  property  is  assessed  at  $7800 
and  who  pays  for  two  polls?  Find  the  collector’s  commission. 

308.  Ten  cylindrical  tanks  of  oil,  each  18  inches  in  diameter  and  30  inches 
long,  inside  measurement,  and  weighing  225  lbs.  each,  were  imported  from  Chris- 
tiana. The  invoice  price  was  10  crowms  per  gallon,  commission  3^%,  consul 
fees  85  crowns,  freight  10c.  per  cwt.,  insurance  2%  ; an  allowance  of  1J%  was 
made  for  leakage.  The  ad  valorem  duty  was  40%  and  the  specific  duty  was 
10J  cents  a gallon.  A draft  to  cover  all  costs  in  Christiana  was  purchased  in 
Philadelphia,  the  exchange  being  .274.  At  what  price  shall  the  oil  be  marked 
per  gallon,  to  fall  20%  of  the  marked  price  and  lose  10%  of  all  in  bad  sales  and 
still  make  a gain  of  25  % ? 

309.  An  importer  of  Philadelphia  owed  on  foreign  invoices  as  follows:  To 

C.  Shepherd  Sons,  London,  £1600  12s. ; -J.  L.  Von  Buesche,  Berlin,  1400  marks ; 
Perrie,  Buzzell  & Co.,  Paris,  3027  francs ; F.  Gonzalez,  Mexico,  715  pesos. 
Exchange  on  London,  4.875;  on  Berlin,  97J ; on  Paris,  5.18J ; on  Mexico,  78J. 
He  purchased  through  his  brokers,  wdio  charged  him  brokerage  ; and  issued 
one  check  to  cover  the  total  cost.  What  was  the  amount  of  the  check  ? 

310.  A commission  merchant  of  San  Francisco  bought  for  a Philadelphia 
firm  a consignment  of  120  chests  of  green  tea,  each  containing  50  lbs.  at  22f 
cents  a pound ; 15  cases  of  Muscatel  raisins,  each  containing  20  boxes  of  50  lbs. 
each,  at  18^  cents  a pound  ; 10  cases  of  California  prunes,  each  containing  20 
boxes  of  25  lbs.  each,  at  11 J cents  a pound;  charges  2%  commission,  3%  guar- 
anty, and  1J%  insurance.  The  firm  in  Philadelphia  remitted  a 30-day  draft  for 
the  amount  due  at  f % premium.  What  was  its  face  and  what  did  the  firm  pay 
for  the  draft,  if  the  rate  of  interest  was  6%  ? 


GENERAL  REVIEW  PROBLEMS 


359 


311.  Find  the  cost,  at  $2.08  a }Tard,  of  carpeting  a room  32J  feet  long,  12J  feet 
wide  and  12  feet  high,  the  carpet  (which  is  22  inches  wide)  being  laid  the  more 
economical  way,  and  one  foot  being  allowed  to  each  strip  for  matching.  The 
room  has  five  windows,  each  7|  feet  by  4J  feet ; three  doors,  each  9J  feet  by  4J  feet ; 
two  fire-places,  each  5 feet  by  6 feet,  and  a baseboard  10  inches  high.  Find  the 
cost  of  papering  the  room  at  50  cents  a double  roll,  and  putting  on  a border 
15  inches  wide,  which  costs  20  cents  a yard.  What  must  be  the  face  of  a 90-day 
note,  dated  and  discounted  to-day,  to  cancel  the  total  cost  ? 

312.  I sent  my  agent  in  London  $10000  with  instructions  to  purchase  cloth 
at  15s.  Qd.  per  yard,  after  paying  consul  fees  of  £15  8s.  and  charging  a commis- 
sion of  3%.  What  amount  is  unexpended,  supposing  he  bought  a whole 
number  of  yards  ? The  duty  was  20  % ad  valorem,  3c.  a yard  specific,  the  freight 
amounted  to  Jd.  a yard,  and  other  charges  to  $125.  After  keeping  the  cloth 
three  months  and  charging  interest  at  6%  on  total  cost,  I sell  it  at  $4.25  a yard 
and  receive  in  payment  two  notes  of  equal  sums,  dated  to-day,  one  at  30  days, 
the  other  at  two  months,  both  of  which  I have  discounted  immediately.  What 
is  my  gain  or  loss  on  the  investment? 

313.  Bought  a consignment  of  wool  in  Buenos  Ayres,  the  gross  weight  of 
which  was  34675  lbs.  @ 15  pesos  per  lb. ; an  allowance  of  8%  of  the  gross  weight 
was  made  for  tare;  freight  charges  were  9f  cents  per  cwt. ; ad  valorem  duty 
18f  % ; specific  duty  3|  cents  per  lb. ; consul’s  fee  85  pesos.  What  was  the  cost 
in  Philadelphia  at  § % premium,  of  a sight  draft  to  cover  all  expenses  in  Buenos 
Ayres  ? At  2 % commission  what  will  my  agent  receive?  At  what  price  per  lb. 
must  it  be  sold  to  make  a clear  profit  of  20%,  after  allowing  5%  for  damaged 
wool,  and  10%  of  my  sales  forbad  debts? 

311/..  A Philadelphia  soap  manufacturer  makes  a cake  of  soap  3 inches  long, 
2 inches  wide  and  J inch  thick.  He  wishes  to  make  a sample  cake  of  the  same 
soap,  keeping  the  proportions  of  the  larger  cake,  but  using  only  one-half  the 
material.  Find  the  dimensions  of  the  sample  size. 

315.  Borrowed  $18360  for  one  year  at  6%.  At  the  end  of  3 months  I 
invested  in  produce  through  an  agent  charging  2%  ; after  keeping  the  goods  4 
months  I sold  one-half  of  them  at  18f  % profit,  and  the  remainder  at  15%  profit, 
paying  2%  commission  for  selling.  I loaned  the  money  I received  at  f%  a 
month  for  the  remainder  of  the  year.  What  was  my  year’s  profit  ? 

316.  A legacy  of  $436.85  was  left  to  a minor ; it  was  invested  at  4 J%  com- 
pound interest,  and  when  the  legatee  wTas  21  years  old  it  amounted  to  $750. 
How  old  was  the  child  when  the  bequest  was  made? 

317.  A public  square  200  yards  on  the  building  line  has  -walks  20  feet  wide 
around  it,  and  two  walks  of  the  same  width  diagonally  across  the  square.  The 
cost  of  paving  is  $1.50  per  square  yard.  For  -what  sum  must  a 60-day  note  be 
drawn  to-day  that  when  discounted  will  pay  the  debt  ? 

318.  A factory  worth  $3000  and  its  contents  are  insured  for  $10000,  as  fol- 
lows : $2000  on  the  building,  $3000  on  machinery  worth  $5000,  and  $5000  on 


360 


GENERAL  REVIEW  PROBLEMS 


stock  worth  $8000.  The  building  is  damaged  by  fire  to  the  amount  of  $1000, 
the  machinery  $4000,  and  the  stock  is  a complete  loss.  What  was  the  premium 
at  ? If  the  risk  is  covered  (1)  by  an  ordinary  policy,  (2)  by  a policy  con- 
taining the  “average  clause,”  (3)  by  a policy  containing  the  80%  co-insurance 
clause,  what  was  the  owner’s  loss  ? What  was  the  company’s  loss? 

319.  What  is  the  difference  in  volume  between  two  wedges,  each  being  60 
inches  in  altitude  and  25  inches  broad  at  the  base,  but  the  base  of  one  being  70 
inches  long  and  its  edge  50  inches,  while  the  base  of  the  other  is  50  inches  long 
and  its  edge  70  inches  ? 

320.  A room  is  35  feet  long,  28  feet  wide  and  12  feet  high.  What  is  the 
shortest  distance  a spider  may  crawl  in  a straight  line  from  a lower  corner  at 
one  end  to  the  opposite  upper  corner  at  the  other  end  ? 


APPENDIX 


THE  METRIC  SYSTEM 


In  1875  a treaty  was  signed  at  Paris  by  seventeen  of  the  principal  nations 
of  the  world,  the  United  States  being  among  the  number,  which  provided  for  the 
permanent  organization  of  an  International  Bureau  of  Weights  and  Measures 
under  the  direction  of  an  International  Committee.  The  most  important  work 
of  the  International  Committee  was  to  provide  for  the  construction  of  a sufficient 
number  of  platinum-iridium  meters  and  kilograms  to  meet  the  demand  of  the 
interested  nations.  The  comparison  of  all  these  standards  with  one  another  and 
with  the  original  meter  and  kilogram  was  made  at  the  International  Bureau 
which  had  been  established  near  Paris  on  neutral  territory  ceded  to  the  Inter- 
national Committee  by  the  French  Government. 

This  work  was  completed  in  1889,  and  after  selecting  a certain  meter  and  a 
certain  kilogram  as  the  international  prototypes,  the  others  were  distributed  by 
lot  to  the  different  countries.  The  international  meter  and  kilogram  have  values 
identical  with  the  original  meter  and  kilogram,  are  preserved  in  a special  under- 
ground vault  at  the  International  Bureau,  and  are  accessible  only  to  the 
International  Committee.  The  United  States  secured  two  meters  and  two  kilo- 
grams, which  are  now  preserved  at  the  Bureau  of  Standards  at  Washington  and 
serve  as  the  fundamental  standards  of  length  and  mass  of  the  United  States.  It 
is  the  plan  of  the  International  Committee  to  intercompare  all  the  national 
meters  and  kilograms  with  the  international  prototypes  at  regular  intervals  or 
whenever  considered  necessary. 

At  the  present  time  the  International  Bureau  of  Weights  and  Measures  is 
supported  jointly  by  the  following  countries:  The  United  States,  (its  use  “ made 
lawful  throughout  the  United  States  ” by  Act  of  Congress  in  1866),  Great  Britain, 
Germany,  Russia,  France,  Austria-Hungary,  Belgium,  Argentine  Confederation, 
Spain,  Italy,  Mexico,  Peru,  Portugal,  Roumania,  Servia,  Sweden,  Norway,  Swit- 
zerland, Venezuela,  Japan,  and  Denmark. 

The  advantages  claimed  for  the  metric  system  are  : 

(1)  The  decimal  relation  between  the  units. 

(2)  The  extremely  simple  relation  of  the  units  of  length,  area,  volume,  and 
weight  to  one  another. 

(3)  The  uniform  and  self-defining  names  of  units. 

Note. — The  facts  contained  in  this  appendix  are  based  on  a pamphlet  issued  by  the  Department 
of  Commerce  and  Labor,  entitled  “ The  International  System  of  Weights  and  Measures.” 


361 


362 


THE  METRIC  SYSTEM 


SYNOPSIS  OF  THE  SYSTEM 

The  fundamental  unit  of  the  metric  system  is  the  Meter — the  unit  of  length. 
From  this  the  units  of  capacity  (Liter)  and  of  weight  (Gram)  were  derived.  All 
other  units  are  the  decimal  subdivisions  or  multiples  of  these.  These  three  units 
are  simply  related  ; e.  g.,  for  all  practical  purposes  one  cubic  decimeter  equals 
one  liter  and  one  liter  of  water  weighs  one  kilogram.  The  metric  tables  are 
formed  by  combining  the  words  “ meter,”  “ gram,”  and  “ liter,”  with  the  six 
numerical  prefixes,  as  in  the  following  tables: 


PREFIXES 

MEANING 

UNITS 

milli- 

one  thousandth 

1 

TOTO 

.001  , 

1 meter,  for  length. 

centi- 

one  hundredth 

1 

10  0 

.01  I 

deci- 

one  tenth 

1 

1 0 

.1 

Unit 

one 

1 

\ gram,  for  weight  or  mass. 
1 

deka- 

ten 

1 0 
1 

10 

hecto- 

one  hundred 

10  0 
] 

100  | 

, liter,  for  capacity. 

kilo- 

one  thousand 

10  0 0 
1 

1000 

MEASURES  OF  LENGTH 


The  meter  is  the  unit  of  length  and  is  used  for  measuring  dry  goods,  mer- 
chandise, engineering  construction,  building  and  other  purposes  where  the  yard 
and  foot  are  used.  The  meter  is  39.37  inches  long,  about  a tenth  longer  than  the 


yard. 


10  millimeters  (mm.) 
10  centimeters 
10  decimeters 
10  meters 
10  dekameters 
10  hectometers 


= 1 centimeter  (cm.) 

= 1 decimeter  (dm.) 

- - 1 meter  (m.) 

= 1 dekameter  (dekam.) 
= 1 hectometer  (hm.) 

= 1 kilometer  (km.) 


The  centimeter  and  millimeter  are  used  in  machine  construction  and  similar 


work,  instead  of  the  inch  and  its  fractions.  The  centimeter,  as  its  name  shows, 
is  the  hundredth  of  a meter.  It  is  used  in  cabinet  work,  in  expressing  sizes  of 
paper,  books,  and  in  many  cases  where  the  inch  is  used.  The  centimeter  is  about 
two-fifths  of  an  inch  and  the  millimeter  about  one  twenty-fifth  of  an  inch.  The 
millimeter  is  divided  for  finer  work  into  tenths,  hundredths  and  thousandths. 


1 1 1 1 1 1 1 1 i iiii|iiii  iiiijim  1 1 1 1 1 1 1 1 1 iiii|iiii  mijim  iiii|im  mijim  iiiijini  iiiijim 
0 1 2 3 4-  5 6 7 8 9 10  cm. 


O 1 2 3 4 m. 


l I I ll  I I I I I I I I I I 


1 1 1!  1 1 1 


1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 


Fiq.  1.  Comparison  Scale:  10  Centimeters  and  4 Inches.  (Actual  Size 


THE  METRIC  SYSTEM 


363 


Where  miles  are  used  in  England  and  the  United  States  for  measuring 
distances,  the  kilometer  (1000  meters)  is  used  in  metric  countries.  The  kilometer 
is  about  5 furlongs.  There  are  about  1600  meters  in  a statute  mile,  20  meters  in 
a chain,  and  5 meters  in  a rod. 

If  a number  of  distances  in  millimeters,  meters  and  kilometers  are  to  be 
added,  reduction  is  not  necessary.  They  are  added  as  dollars,  dimes  and  cents 
are  now  added.  For  example,  “1050.25  meters”  is  not  read  “1  kilometer,  5 
dekameters  and  5 centimeters,”  but  “ one  thousand  fifty  meters,  twenty-five 
centimeters,”  just  as  “ $1050.25  ” is  read  “ one  thousand  fifty  dollars  and  twenty- 
five  cents.” 

SURFACE  MEASUREMENT 


The  table  of  areas  is  formed  by  squaring  the  length  measures,  as  in  our 
common  system.  For  land  measure  10  meters  square  is  called  an  “ Are  ” (meaning 
“ area  ”).  The  side  of  one  “ are  ” is  about  33  feet. 

TABLE 

10  milliares  (ma.)  = 1 centiare  (ca.) 


10  centiares 
10  deciares 
10  ares 
10  dekares 
10  hectares 


= 1 deciare  (da.) 
= 1 are  (A.  or  a.) 

- 1 dekare 
— 1 hectare  (ha.) 
= 1 kilare  (ka.) 


The  Hectare  is  100  meters  square,  and,  as  its  name  indicates,  is  100  ares, 
or  about  2J  acres.  An  acre  is  about  0.4  hectare.  A standard  United  States 
quarter  section  contains  almost  exactly  64  hectares.  A square  kilometer  contains 
100  hectares. 

For  smaller  measures  of  surface  the  square  meter  is  used.  The  square  meter 
is  about  20  per  cent,  larger  than  the  square  yard.  For  still  smaller  surfaces  the 
square  centimeter  is  used.  A square  inch  contains  about  6|  square  centimeters. 


MEASURES  OF  VOLUMES 

The  cubic  measures  are  the  cubes  of  the  linear  units.  The  cubic  meter 
(sometimes  called  the  stere,  meaning  “ solid  ”)  is  the  unit  of  volume. 

It  is  used  in  place  of  the  cubic  yard  and  is  about  30  per  cent,  larger.  This 
is  used  for  “cuts  and  fills”  in  grading  land,  measuring  timber,  expressing 
contents  of  tanks  and  reservoirs,  flow  of  rivers,  dimensions  of  stone,  tonnage  of 
ships,  and  other  places  where  the  cubic  yard  and  foot  are  used. 

TABLE 

1000  cubic  millimeters  (mm.3)  = 1 cubic  centimeter  (c.c.  or  cm.3) 

1000  cubic  centimeters  = 1 cubic  decimeter  (dm.3) 

1000  cubic  decimeters  = 1 cubic  meter  (in.3  or  stere) 

The  thousandth  part  of  the  cubic  meter  (1  cubic  decimeter)  is  called  the 
Liter.  (See  table  of  capacity  units.) 

For  very  small  volumes  the  cubic  centimeter  is  used.  This  volume  of 
water  weighs  a gram,  which  is  the  unit  of  weight  or  mass.  There  are  about 


364 


THE  METRIC  SYSTEM 


16  cubic  centimeters  in  a cubic  inch.  The  cubic  centimeter  is  the  unit  of  volume 
used  by  chemists  as  well  as  in  pharmacy,  medicine,  surgery,  and  other  technical 
work.  One  thousand  cubic  centimeters  make  1 liter.  A cubic  meter  of  water 
weighs  a metric  ton  and  is  equal  to  1 kiloliter. 


MEASURES  OF  CAPACITY 


The  Liter  is  the  unit  of  capacity.  It  equals  a cubic  decimeter. 

The  liter  is  used  for  measurements  commonly  given  in  the  gallon,  the  liquid 
and  dry  quarts,  a liter  being  5 per  cent,  larger  than  our  liquid  quart  and  10  per 
cent,  smaller  than  the  dry  quart. 


TABLE 

10  milliliters  (ml.)  = 1 centiliter  (cl.) 

10  centiliters  = 1 deciliter  (dl.) 

10  deciliters  = 1 liter  (1.) 

10  liters  = 1 dekaliter  (dekal.) 

10  dekaliters  = 1 hectoliter  (hi.) 

10  hectoliters  = 1 kiloliter  (kl.) 


Fig  3 

comparison  of  the  dry  quart.  ut£r 

Ar«0  LIQUID  QUART.  I ACTUAL  SlZE-1 


366 


THE  METRIC  SYSTEM 


The  hectoliter  (100  liters)  serves  the  same  purpose  as  the  United  States 
bushel  (2150.4  cubic  inches),  and  is  equal  to  about  3 bushels,  or  a barrel.  A peck 
is  about  9 liters. 

A liter  of  water  weighs  exactly  a kilogram,  i.  e.,  1000  grams.  A thousand 
liters  of  water  weigh  1 metric  ton. 


MEASURES  OF  WEIGHT 

The  Gram  is  the  unit  of  weight.  It  is  the  weight  of  a cubic  centimeter  of 
distilled  water  at  freezing  temperature,  and  weighs  15.432  Troy  grains. 


TABLE 


10  milligrams  (mg( 
10  centigrams 
10  decigrams 
10  grams 
1 0 dekagrams 
10  hectograms 


= 1 centigram  (eg.) 

= 1 decigram  (dg.) 

= 1 Sram  (g-) 

= 1 dekagram  (dekag.) 
= 1 hectogram  (hg.) 

= 1 kilogram  (kg.) 


Measurements  commonly  expressed  in  gross  tons  or  short  tons  are  stated  in 
metric  tons  (1000  kilograms).  The  metric  ton  comes  between  our  long  and  short 
tons  and  serves  the  purpose  of  both. 


The  kilogram  and  “ half  kilo  ” serve  for  every-day  trade,  the  latter  being  10 
per  cent,  larger  than  the  pound.  The  kilogram  is  approximately  2.2  pounds. 


THE  METRIC  SYSTEM 


367 


Fig.  5.  Relative  Size  of  Avoirdupois  Ounce,  30-Gram, 
and  Troy  Ounce  (Brass)  Weights.  (Actual  Size.) 


• 0 0 

Fig.  6.  Relative  Size  of 
Gram  and  Scruple 
»Brass)  Weights. 
(Actual  Size.) 


The  gram  and  its  multiples  and  divisions  are  used  for  the  same  purposes  as 
ounces,  pennyweights,  drams,  scruples  and  grains.  For  foreign  postage,  30 
grams  is  the  legal  equivalent  of  the  avordupois  ounce. 


Duke  University  Libraries 

.. 

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511  L957  223C32 


